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Plane Geometry

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Question 101Question →

A convex polygon has nn sides. The interior angles of the polygon form an arithmetic progression with a common difference of d∘d^\circ, where dd is a positive integer. If the smallest interior angle of the polygon measures 100∘100^\circ, what is the maximum possible value of nn?

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Answer: 8

Answer

The maximum possible value of nn is 8.
The correct answer is 8 because we set the sum of the interior angles (n−2)×180∘(n-2) \times 180^\circ equal to the sum of the arithmetic progression n2[200+(n−1)d]\frac{n}{2}[200 + (n-1)d]. Solving for (n−1)d(n-1)d gives (n−1)d=160−720n(n-1)d = 160 - \frac{720}{n}. Since the polygon is convex, the largest angle 100+(n−1)d100 + (n-1)d must be strictly less than 180∘180^\circ, which means (n−1)d<80∘(n-1)d < 80^\circ. Substituting the expression yields 160−720n<80160 - \frac{720}{n} < 80, which simplifies to n<9n < 9. Since nn must be an integer, the maximum possible value of nn is 8. For n=8n = 8, the common difference d=10d = 10 is a positive integer, satisfying all conditions.

Step-by-Step Solution

1
Express the sum of the interior angles using the polygon angle sum formula and the arithmetic progression formula.
The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. The sum of the angles in arithmetic progression with first term 100∘100^\circ and common difference d∘d^\circ is n2[2(100)+(n−1)d]=100n+n(n−1)d2\frac{n}{2}[2(100) + (n-1)d] = 100n + \frac{n(n-1)d}{2}.
This sets up the system relating the geometry of the polygon to the algebraic progression of its angles.
2
Equate the two expressions and solve for the quantity (n−1)d(n-1)d.
100n+n(n−1)d2=180n−360  ⟹  n(n−1)d2=80n−360  ⟹  (n−1)d=160−720n100n + \frac{n(n-1)d}{2} = 180n - 360 \implies \frac{n(n-1)d}{2} = 80n - 360 \implies (n-1)d = 160 - \frac{720}{n}.
This isolates the quantity (n−1)d(n-1)d, which represents the difference between the largest and smallest angles.
3
Apply the convexity constraint that every interior angle must be strictly less than 180∘180^\circ.
The largest angle is the last term of the progression: 100+(n−1)d100 + (n-1)d. For the polygon to be convex, this angle must be strictly less than 180∘180^\circ. Therefore, 100+(n−1)d<180  ⟹  (n−1)d<80100 + (n-1)d < 180 \implies (n-1)d < 80. Substituting (n−1)d=160−720n(n-1)d = 160 - \frac{720}{n} gives 160−720n<80  ⟹  80<720n  ⟹  n<9160 - \frac{720}{n} < 80 \implies 80 < \frac{720}{n} \implies n < 9.
A convex polygon cannot have any interior angles greater than or equal to 180∘180^\circ.
4
Identify the maximum integer value of nn and verify that a positive integer common difference dd exists.
Since n<9n < 9 and nn must be an integer, the maximum possible value is n=8n = 8. For n=8n = 8, we calculate (8−1)d=160−7208  ⟹  7d=70  ⟹  d=10(8-1)d = 160 - \frac{720}{8} \implies 7d = 70 \implies d = 10. Since d=10d = 10 is a positive integer, the solution is valid.
This ensures the result satisfies all constraints, including that the common difference is a positive integer.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and all interior angles of a convex polygon must be strictly less than 180∘180^\circ.
Question 102Question →

An irregular convex octagon has five interior angles that each measure 144∘144^\circ. The remaining three interior angles have measures in the ratio 3:4:53:4:5. What is the measure, in degrees, of the largest interior angle of this octagon?

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Answer: 150

Answer

The measure of the largest interior angle of the octagon is 150∘150^\circ.
The total sum of the interior angles of an octagon is (8−2)×180∘=1080∘(8-2) \times 180^\circ = 1080^\circ. The sum of the five given angles is 5×144∘=720∘5 \times 144^\circ = 720^\circ, leaving a sum of 1080∘−720∘=360∘1080^\circ - 720^\circ = 360^\circ for the remaining three angles. Since these three angles are in the ratio 3:4:53:4:5, we set 3x+4x+5x=360∘3x + 4x + 5x = 360^\circ, yielding 12x=360∘12x = 360^\circ and x=30∘x = 30^\circ. The largest of these three angles is 5×30∘=150∘5 \times 30^\circ = 150^\circ. Comparing 150∘150^\circ to the other angles of the octagon (which are 144∘144^\circ, 90∘90^\circ, and 120∘120^\circ), the largest interior angle is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of the octagon.
The total sum is 1080∘1080^\circ.
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n - 2) \times 180^\circ. For an octagon, n=8n = 8, so the sum is (8−2)×180∘=6×180∘=1080∘(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ.
2
Calculate the sum of the five given congruent angles.
The sum of these five angles is 720∘720^\circ.
Since five angles each measure 144∘144^\circ, their combined sum is 5×144∘=720∘5 \times 144^\circ = 720^\circ.
3
Determine the sum of the remaining three interior angles.
The sum of the remaining angles is 360∘360^\circ.
Subtracting the sum of the five congruent angles from the total sum of the octagon's interior angles yields the sum of the remaining three angles: 1080∘−720∘=360∘1080^\circ - 720^\circ = 360^\circ.
4
Use the ratio 3:4:53:4:5 to find the measures of the remaining three angles.
The measures of the three angles are 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ.
Let the measures of the remaining three angles be 3x3x, 4x4x, and 5x5x. Their sum is 3x+4x+5x=12x=360∘3x + 4x + 5x = 12x = 360^\circ, which gives x=30∘x = 30^\circ. The largest of these three angles is 5×30∘=150∘5 \times 30^\circ = 150^\circ.
5
Compare the measures of all interior angles of the octagon to find the largest one.
The largest angle is 150∘150^\circ.
The octagon's interior angles consist of five angles of 144∘144^\circ, and three angles of 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. Comparing these values, 150∘150^\circ is the largest measure.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratios can be used to partition a total sum into specific parts.
Estimated Time:2m 0s
Question 103Question →

In △ABC\triangle ABC, the measure of ∠A\angle A is 40∘40^\circ. The measure of ∠B\angle B is three times the measure of ∠C\angle C. What is the measure, in degrees, of ∠B\angle B?

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Answer: 105

Answer

The measure of angle B is 105 degrees.
The sum of the interior angles in any triangle is 180∘180^\circ. By setting the measure of ∠C\angle C to xx and the measure of ∠B\angle B to 3x3x, we can write the equation 40+3x+x=18040 + 3x + x = 180. Solving for xx yields 4x=1404x = 140, which simplifies to x=35x = 35. The measure of ∠B\angle B is 3x3x, which is 3×35=105∘3 \times 35 = 105^\circ.

Step-by-Step Solution

1
Set up the equation using the fact that the sum of angles in a triangle is 180 degrees.
m∠A+m∠B+m∠C=180∘m\angle A + m\angle B + m\angle C = 180^\circ
The angles of any triangle in plane geometry sum to 180 degrees.
2
Represent the unknown angles algebraically.
Let m∠C=xm\angle C = x, then m∠B=3xm\angle B = 3x. Substitute m∠A=40∘m\angle A = 40^\circ.
This allows solving for the unknown angles with a single-variable equation.
3
Solve the equation for xx.
40+4x=180  ⟹  4x=140  ⟹  x=3540 + 4x = 180 \implies 4x = 140 \implies x = 35
To find the measure of angle C.
4
Calculate the measure of angle B.
m∠B=3(35)=105∘m\angle B = 3(35) = 105^\circ
Angle B is three times angle C, and we need to find the measure of angle B.

Key Concept

The sum of the interior angles of a triangle is always 180 degrees.
Question 104Question →

In right triangle ABCABC, the measure of ∠B\angle B is 90∘90^\circ, and the measure of ∠C\angle C is 30∘30^\circ. Point DD lies on segment BCBC such that the measure of ∠ADC\angle ADC is 135∘135^\circ. If the length of segment ADAD is 88 units, what is the length, in units, of segment ACAC?

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Answer: 828\sqrt{2}

Answer

828\sqrt{2}
The correct answer is 828\sqrt{2}. Since points BB, DD, and CC lie on a straight line, the adjacent angles ∠ADB\angle ADB and ∠ADC\angle ADC must sum to 180∘180^\circ. Subtracting the given angle measure shows that ∠ADB=180∘−135∘=45∘\angle ADB = 180^\circ - 135^\circ = 45^\circ. In the right triangle ABDABD, since ∠B=90∘\angle B = 90^\circ and ∠ADB=45∘\angle ADB = 45^\circ, the triangle is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ special right triangle. Using the ratio of side lengths for this triangle type, the leg ABAB is equal to the hypotenuse ADAD divided by 2\sqrt{2}, which simplifies to AB=82=42AB = \frac{8}{\sqrt{2}} = 4\sqrt{2}. Next, looking at the larger right triangle ABCABC, the angle at CC is given as 30∘30^\circ, which makes △ABC\triangle ABC a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle. In this type of triangle, the hypotenuse ACAC is twice the length of the shorter leg ABAB, which is opposite the 30∘30^\circ angle. Multiplying the length of ABAB by 22 yields AC=2×42=82AC = 2 \times 4\sqrt{2} = 8\sqrt{2} units.

Step-by-Step Solution

1
Find the measure of angle ADBADB using the supplementary angle relationship along the line segment BCBC.
∠ADB=180∘−135∘=45∘\angle ADB = 180^\circ - 135^\circ = 45^\circ
Points BB, DD, and CC are collinear, meaning ∠ADB\angle ADB and ∠ADC\angle ADC form a linear pair and must sum to 180∘180^\circ.
2
Determine the properties of right triangle ABDABD and solve for the length of side ABAB.
△ABD\triangle ABD is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle, where leg AB=AD2=82=42AB = \frac{AD}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 4\sqrt{2} units.
Since ∠B=90∘\angle B = 90^\circ and ∠ADB=45∘\angle ADB = 45^\circ, the remaining angle ∠DAB\angle DAB is also 45∘45^\circ. In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle, the leg length equals the hypotenuse divided by 2\sqrt{2}.
3
Use the properties of the larger 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle ABCABC to find the length of hypotenuse ACAC.
AC=2×AB=2×42=82AC = 2 \times AB = 2 \times 4\sqrt{2} = 8\sqrt{2} units.
In right triangle ABCABC, the angle opposite leg ABAB is ∠C=30∘\angle C = 30^\circ. In any 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle, the hypotenuse is exactly twice the length of the leg opposite the 30∘30^\circ angle.

Key Concept

Using multi-step properties of special right triangles (45∘−45∘−90∘45^\circ-45^\circ-90^\circ and 30∘−60∘−90∘30^\circ-60^\circ-90^\circ) sharing a common boundary line.
Question 105Question →

The measures of the five exterior angles of a convex pentagon are in the ratio 2:3:4:4:52:3:4:4:5. What is the measure of the largest interior angle of this pentagon?

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Answer: 140∘140^\circ

Answer

The measure of the largest interior angle of the pentagon is 140∘140^\circ.
The correct answer is 140∘140^\circ. The sum of the exterior angles of any convex polygon is 360∘360^\circ. For a pentagon with exterior angles in the ratio 2:3:4:4:52:3:4:4:5, the sum of the ratio parts is 2+3+4+4+5=182+3+4+4+5=18 parts. Each part is equal to 360∘/18=20∘360^\circ / 18 = 20^\circ. The smallest exterior angle has 22 parts, which measures 2×20∘=40∘2 \times 20^\circ = 40^\circ. Since each interior angle is supplementary to its corresponding exterior angle, the largest interior angle corresponds to the smallest exterior angle. Thus, the largest interior angle measures 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.

Step-by-Step Solution

1
Determine the sum of the exterior angles of a convex pentagon.
The sum of the exterior angles of any convex polygon is 360∘360^\circ.
This is a fundamental property of convex polygons and provides the total value to distribute among the ratio parts.
2
Calculate the measure of the smallest exterior angle using the given ratio of 2:3:4:4:52:3:4:4:5.
The sum of the ratio parts is 2+3+4+4+5=182+3+4+4+5=18. The smallest exterior angle has 22 parts, so its measure is 218×360∘=40∘\frac{2}{18} \times 360^\circ = 40^\circ.
The largest interior angle will be adjacent (and supplementary) to the smallest exterior angle.
3
Calculate the measure of the largest interior angle by subtracting the smallest exterior angle from 180∘180^\circ.
The largest interior angle is 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ.
An interior angle and its adjacent exterior angle are supplementary and add up to 180∘180^\circ.

Key Concept

The sum of the exterior angles of any convex polygon is 360∘360^\circ. An interior angle and its adjacent exterior angle are supplementary, meaning they add up to 180∘180^\circ. Consequently, the largest interior angle corresponds to the smallest exterior angle.
Estimated Time:1m 30s
Question 106Question →

An irregular convex hexagon has two interior angles that are right angles. The remaining four interior angles have measures in the ratio 4:5:5:64:5:5:6. What is the measure, in degrees, of the largest interior angle of this hexagon?

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Answer: 162

Answer

The correct answer is 162162 degrees. The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the two right angles (180∘180^\circ total) leaves a remaining sum of 540∘540^\circ. The ratio of the remaining four angles is 4:5:5:64:5:5:6, which can be represented as 4x,5x,5x,6x4x, 5x, 5x, 6x, summing to 20x=54020x = 540. Solving for the multiplier gives x=27x = 27. The largest angle is 6(27)=162∘6(27) = 162^\circ, which is also greater than the two 90∘90^\circ angles.
The sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the two right angles (180∘180^\circ) gives a remaining sum of 540∘540^\circ for the other four angles. Let these four angles be 4x,5x,5x,4x, 5x, 5x, and 6x6x. Their sum is 20x=54020x = 540, which solves to x=27x = 27. The largest angle is 6x=6(27)=162∘6x = 6(27) = 162^\circ, which is also larger than the two 90∘90^\circ angles.

Step-by-Step Solution

1
Calculate the sum of all interior angles of a convex hexagon.
The sum is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ.
The sum of the interior angles of any convex nn-gon is given by (n−2)×180∘(n-2) \times 180^\circ.
2
Subtract the sum of the two right angles from the total sum.
The remaining sum is 720∘−180∘=540∘720^\circ - 180^\circ = 540^\circ.
Two right angles contribute 90∘+90∘=180∘90^\circ + 90^\circ = 180^\circ to the total.
3
Set up a linear equation representing the ratio of the remaining four angles.
The equation is 4x+5x+5x+6x=5404x + 5x + 5x + 6x = 540, which simplifies to 20x=54020x = 540, yielding x=27x = 27.
The angles are proportional to the parts of the ratio, and their sum must equal the remaining 540∘540^\circ.
4
Calculate the largest angle from the ratio and compare with the right angles.
The largest angle is 6×27=162∘6 \times 27 = 162^\circ.
The largest term in the ratio is 66, and the resulting angle 162∘162^\circ is larger than both 90∘90^\circ and the other calculated angles (108∘108^\circ and 135∘135^\circ).

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratio relationships can be solved using algebraic multipliers.
Question 107Question →

An irregular convex decagon (10-sided polygon) has four interior angles that each measure 150∘150^\circ. The remaining six interior angles are congruent to each other. What is the degree measure of each of these remaining six angles?

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Answer: 140

Answer

The measure of each of the remaining six interior angles is 140∘140^\circ.
The total sum of the interior angles of a 10-sided convex polygon is (10−2)×180∘=1,440∘(10-2) \times 180^\circ = 1,440^\circ. Subtracting the sum of the four angles that each measure 150∘150^\circ (4×150∘=600∘4 \times 150^\circ = 600^\circ) leaves 840∘840^\circ for the remaining six angles. Since these remaining six angles are congruent, each measures 840∘÷6=140∘840^\circ \div 6 = 140^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of a convex decagon.
1,440∘1,440^\circ
The interior angle sum of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and a decagon has 10 sides.
2
Find the sum of the four angles measuring 150∘150^\circ.
600∘600^\circ
Multiply the number of angles by their given degree measure.
3
Determine the sum of the remaining six congruent angles.
840∘840^\circ
Subtract the sum of the four known angles from the total interior angle sum of the decagon.
4
Divide the remaining sum by the number of congruent angles.
140∘140^\circ
Since the remaining six angles are equal in measure, dividing their sum by 6 yields the measure of each individual angle.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ.
Question 108Question →

An irregular convex hexagon has three interior angles that each measure 110∘110^\circ. The remaining three interior angles have measures in the ratio 3:4:53:4:5. What is the degree measure of the exterior angle corresponding to the smallest interior angle of this hexagon?

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Answer: 82.5∘82.5^\circ

Answer

82.5∘82.5^\circ
The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the three known angles (330∘330^\circ) leaves 390∘390^\circ for the remaining three angles. Using the ratio 3:4:53:4:5, the smallest of these remaining angles is calculated as 312×390∘=97.5∘\frac{3}{12} \times 390^\circ = 97.5^\circ. Since 97.5∘97.5^\circ is smaller than the other angles (110∘110^\circ, 130∘130^\circ, 162.5∘162.5^\circ), it is the smallest interior angle of the hexagon. The exterior angle is supplementary to the interior angle, so it measures 180∘−97.5∘=82.5∘180^\circ - 97.5^\circ = 82.5^\circ.

Step-by-Step Solution

1
Calculate the sum of all interior angles of the hexagon.
The sum is (6−2)×180∘=4×180∘=720∘(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
This establishes the total measure of the interior angles of a hexagon.
2
Subtract the sum of the three known interior angles from the total sum.
The remaining sum is 720∘−(3×110∘)=720∘−330∘=390∘720^\circ - (3 \times 110^\circ) = 720^\circ - 330^\circ = 390^\circ.
This determines the combined measure of the three remaining angles.
3
Determine the measures of the remaining three angles using the ratio.
Setting up the equation 3y+4y+5y=390∘3y + 4y + 5y = 390^\circ gives 12y=390∘12y = 390^\circ, or y=32.5∘y = 32.5^\circ. The individual angles are 3×32.5∘=97.5∘3 \times 32.5^\circ = 97.5^\circ, 4×32.5∘=130∘4 \times 32.5^\circ = 130^\circ, and 5×32.5∘=162.5∘5 \times 32.5^\circ = 162.5^\circ.
This finds each of the remaining interior angle measures.
4
Find the smallest interior angle and compute its supplementary exterior angle.
The smallest interior angle of the hexagon is 97.5∘97.5^\circ. The corresponding exterior angle is 180∘−97.5∘=82.5∘180^\circ - 97.5^\circ = 82.5^\circ.
An interior angle and its adjacent exterior angle sum to 180∘180^\circ.

Key Concept

Polygon Angles and Properties
Question 109Question →

A wheelchair ramp is constructed in two consecutive straight segments. The first segment rises at a 30∘30^\circ angle relative to the flat ground and has a length of 1212 feet. The second segment starts at the end of the first segment and rises at a 45∘45^\circ angle relative to the horizontal, with a length of 828\sqrt{2} feet. What is the total vertical rise, in feet, from the start of the first segment to the end of the second segment?

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Answer: 1414

Answer

The total vertical rise is 1414 feet.
The total vertical rise is the sum of the vertical rises of the two individual segments. The first segment forms a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle where the hypotenuse is 1212 feet, so the vertical rise (the leg opposite the 30∘30^\circ angle) is half of the hypotenuse, which is 66 feet. The second segment forms a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle where the hypotenuse is 828\sqrt{2} feet, so the vertical rise (the leg opposite the 45∘45^\circ angle) is the hypotenuse divided by 2\sqrt{2}, which is 88 feet. Adding these two values gives a total vertical rise of 1414 feet.

Step-by-Step Solution

1
Calculate the vertical rise of the first segment.
The first segment has a length of 1212 feet and rises at a 30∘30^\circ angle. It forms a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle where the vertical rise is the side opposite the 30∘30^\circ angle. Since the leg opposite the 30∘30^\circ angle is half the hypotenuse, the vertical rise is 122=6\frac{12}{2} = 6 feet.
To find the vertical component of the first ramp segment.
2
Calculate the vertical rise of the second segment.
The second segment has a length of 828\sqrt{2} feet and rises at a 45∘45^\circ angle. It forms a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle where the leg length is the hypotenuse divided by 2\sqrt{2}. Thus, the vertical rise is 822=8\frac{8\sqrt{2}}{\sqrt{2}} = 8 feet.
To find the vertical component of the second ramp segment.
3
Sum the vertical rises of both segments.
The total vertical rise is 6 feet+8 feet=146\text{ feet} + 8\text{ feet} = 14 feet.
To find the combined vertical height gained over the entire ramp.

Key Concept

Using special right triangle ratios (30∘−60∘−90∘30^\circ-60^\circ-90^\circ and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ) to find missing side lengths.
Estimated Time:1m 30s
Question 110Question →

An equilateral triangle ABCABC has a side length of 88 inches. Point DD lies on side BCBC such that the distance from BB to DD is 33 inches. What is the length, in inches, of the segment ADAD?

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Answer: 7

Answer

The length of segment ADAD is 77 inches.
Dropping altitude AMAM from AA to BCBC divides the equilateral triangle into two 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangles. Since MM is the midpoint of BCBC, BM=4BM = 4 inches. In △ABM\triangle ABM, the hypotenuse is 88 and the shorter leg is 44, so the altitude AM=43AM = 4\sqrt{3} inches. Since BD=3BD = 3 inches, the segment DMDM has length BM−BD=4−3=1BM - BD = 4 - 3 = 1 inch. Applying the Pythagorean Theorem to right triangle △ADM\triangle ADM gives AD2=AM2+DM2=(43)2+12=48+1=49AD^2 = AM^2 + DM^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49, which simplifies to AD=7AD = 7 inches.

Step-by-Step Solution

1
Find the midpoint of side BCBC by dropping altitude AMAM.
BM=4BM = 4 inches
In an equilateral triangle, the altitude to a side bisects that side.
2
Calculate the length of the altitude AMAM.
AM=43AM = 4\sqrt{3} inches
The altitude forms a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle with the hypotenuse of 88 inches, making the altitude length equal to 8×32=438 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}.
3
Determine the length of the segment DMDM.
DM=1DM = 1 inch
Since DD is 33 inches from BB and MM is 44 inches from BB, the remaining distance is 4−3=14 - 3 = 1.
4
Apply the Pythagorean Theorem on right triangle △ADM\triangle ADM to find ADAD.
AD=7AD = 7 inches
The hypotenuse squared is the sum of the squares of the legs: AD2=(43)2+12=48+1=49AD^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49, which gives AD=7AD = 7.

Key Concept

Using the altitude of an equilateral triangle to create special right triangles and applying the Pythagorean Theorem.
Question 111Question →

A rectangular park ABCDABCD has a length of 2020 meters and a width of 1515 meters. A straight walking path is built from corner AA to a point PP on the diagonal path BDBD such that the path APAP is perpendicular to BDBD. What is the length, in meters, of the path APAP?

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Answer: 12

Answer

The length of the path APAP is 1212 meters.
The diagonal BDBD divides the rectangular park into two congruent right triangles. For right triangle ABDABD, the legs are AB=20AB = 20 and AD=15AD = 15. Using the Pythagorean theorem, the hypotenuse BD=202+152=25BD = \sqrt{20^2 + 15^2} = 25 meters. The area of triangle ABDABD is 12×20×15=150\frac{1}{2} \times 20 \times 15 = 150 square meters. Since APAP is perpendicular to BDBD, APAP is the altitude to base BDBD, so the area can also be written as 12×25×AP\frac{1}{2} \times 25 \times AP. Equating the two areas, 12.5×AP=15012.5 \times AP = 150, which simplifies to AP=12AP = 12 meters.

Step-by-Step Solution

1
Calculate the length of the diagonal BDBD using the Pythagorean theorem on right triangle ABDABD.
BD=202+152=25BD = \sqrt{20^2 + 15^2} = 25 meters.
The diagonal forms the hypotenuse of the right triangle ABDABD, which is needed to calculate the altitude APAP.
2
Express the area of right triangle ABDABD using the two perpendicular legs, ABAB and ADAD.
Area=12×20×15=150\text{Area} = \frac{1}{2} \times 20 \times 15 = 150 square meters.
This establishes the total area of the triangle.
3
Express the area of the same triangle using the diagonal BDBD as the base and the perpendicular path APAP as the height.
Area=12×25×AP\text{Area} = \frac{1}{2} \times 25 \times AP.
This sets up an equation using the unknown path length APAP.
4
Equate the two area expressions and solve for APAP.
12.5×AP=150  ⟹  AP=1212.5 \times AP = 150 \implies AP = 12 meters.
Since both expressions represent the area of the same triangle, they must be equal.

Key Concept

Using the Pythagorean theorem to find the hypotenuse of a right triangle, and then using the area formula to find the altitude to the hypotenuse.

Alternative Method

Alternatively, you can use similar right triangles. Since triangle ABPABP is similar to triangle DBADBA, the ratio of their corresponding sides is equal: APAD=ABBD\frac{AP}{AD} = \frac{AB}{BD}. Substituting the known values gives AP15=2025\frac{AP}{15} = \frac{20}{25}, which simplifies to AP=15×45=12AP = 15 \times \frac{4}{5} = 12 meters.
Estimated Time:1m 30s
Question 112Question →

A convex pentagon has one interior angle that measures 108∘108^\circ. The remaining four interior angles have measures in the ratio 3:4:5:63:4:5:6. What is the measure of the largest interior angle of this pentagon?

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Answer: 144∘144^\circ

Answer

The measure of the largest interior angle of the pentagon is 144∘144^\circ.
The correct answer is the option stating 144∘144^\circ. To find this, we calculate the sum of all interior angles of a pentagon using (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Subtracting the known angle of 108∘108^\circ leaves 432∘432^\circ for the remaining four angles. The ratio 3:4:5:63:4:5:6 implies these angles can be represented as 3x3x, 4x4x, 5x5x, and 6x6x, summing to 18x18x. Solving 18x=43218x = 432 gives x=24x = 24. The largest angle is 6x=6×24∘=144∘6x = 6 \times 24^\circ = 144^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a pentagon.
The sum of the interior angles of any convex pentagon (n=5n = 5) is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
Before finding individual angles, we must determine the total sum of all interior angles in the polygon.
2
Determine the sum of the remaining four angles.
The remaining four angles sum to 540∘−108∘=432∘540^\circ - 108^\circ = 432^\circ.
We subtract the measure of the known angle from the total sum to find the portion allocated to the remaining angles.
3
Set up and solve an algebraic equation using the given ratio.
Let the four remaining angles be represented as 3x3x, 4x4x, 5x5x, and 6x6x. Their sum is 3x+4x+5x+6x=18x3x + 4x + 5x + 6x = 18x. Setting this equal to the remaining sum gives 18x=43218x = 432, which simplifies to x=24x = 24.
Using the ratio allows us to define the relative sizes of the remaining angles in terms of a single variable, which we can solve for using their sum.
4
Find the measure of the largest interior angle.
The largest of these four angles is represented by 6x6x, which is 6×24∘=144∘6 \times 24^\circ = 144^\circ. Since 144∘144^\circ is also larger than the given angle of 108∘108^\circ, it is the largest interior angle of the pentagon.
We multiply the value of xx by the largest coefficient in the ratio and compare it to the other given angle to identify the maximum measure.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. For an irregular polygon, when some angles are known and others are in a ratio, we subtract the known angles from the total sum and distribute the remaining sum proportionally according to the ratio.
Estimated Time:1m 30s
Question 113Question →

A circular pizza with a radius of 8 inches8\text{ inches} is cut into slices. If one slice has a central angle of 45∘45^\circ, what is the area, in square inches, of this slice?

Show answer & explanation

Answer: 8π8\pi

Answer

The area of the slice is 8π8\pi square inches.
The area of a sector is found by multiplying the circle's total area by the fraction of the circle that the sector represents. With a radius of 88 inches, the total area is π×82=64π\pi \times 8^2 = 64\pi square inches. Since the slice has a central angle of 45∘45^\circ, it represents 45360=18\frac{45}{360} = \frac{1}{8} of the entire pizza. Multiplying the total area by this fraction gives 18×64π=8π\frac{1}{8} \times 64\pi = 8\pi square inches.

Step-by-Step Solution

1
Calculate the total area of the circular pizza using the area formula A=πr2A = \pi r^2 with r=8r = 8.
The total area of the pizza is π×82=64π\pi \times 8^2 = 64\pi square inches.
To find the area of a sector, we first need to determine the area of the entire circle.
2
Find the fraction of the circle represented by the slice's central angle by dividing 45∘45^\circ by 360∘360^\circ.
The fraction is 45360=18\frac{45}{360} = \frac{1}{8}.
A full circle has 360∘360^\circ, so the ratio of the central angle to 360∘360^\circ gives the proportion of the circle's total area that the sector occupies.
3
Multiply the total area of the pizza by the fraction representing the slice.
The area of the slice is 18×64π=8π\frac{1}{8} \times 64\pi = 8\pi square inches.
Multiplying the total circle area by the sector's fraction yields the sector's area.

Key Concept

The area of a sector is proportional to its central angle and can be found using the formula A=θ360πr2A = \frac{\theta}{360} \pi r^2 when the angle is in degrees.
Question 114Question →

In the figure below, quadrilateral ABCDABCD is divided by diagonal BDBD into two right triangles. In right triangle ABDABD, the angle at BB is a right angle, ∠ADB=30∘\angle ADB = 30^\circ, and the hypotenuse AD=12AD = 12 centimeters. In right triangle BCDBCD, the angle at CC is a right angle, and BC=CDBC = CD. What is the perimeter, in centimeters, of quadrilateral ABCDABCD?

Show answer & explanation

Answer: 18+6618 + 6\sqrt{6}

Answer

The perimeter of the quadrilateral is 18+6618 + 6\sqrt{6} centimeters.
The perimeter of quadrilateral ABCDABCD is the sum of the lengths of its four outer boundary sides: ABAB, BCBC, CDCD, and DADA. In the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle ABDABD, the shorter leg ABAB is half the hypotenuse ADAD, so AB=6 cmAB = 6\text{ cm}. The longer leg BDBD is AB3=63 cmAB\sqrt{3} = 6\sqrt{3}\text{ cm}. In the isosceles right triangle BCDBCD, the hypotenuse is BD=63 cmBD = 6\sqrt{3}\text{ cm}. The legs BCBC and CDCD are congruent, with each length equal to the hypotenuse divided by 2\sqrt{2}, which simplifies to 36 cm3\sqrt{6}\text{ cm}. Adding the four outer side lengths (6+36+36+126 + 3\sqrt{6} + 3\sqrt{6} + 12) yields a perimeter of 18+66 cm18 + 6\sqrt{6}\text{ cm}.

Step-by-Step Solution

1
Find the lengths of the legs of right triangle ABDABD using the properties of a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle.
AB=6 cmAB = 6\text{ cm} and BD=63 cmBD = 6\sqrt{3}\text{ cm}
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the leg opposite the 30∘30^\circ angle is half the length of the hypotenuse (AB=122=6AB = \frac{12}{2} = 6), and the leg opposite the 60∘60^\circ angle is 3\sqrt{3} times the shorter leg (BD=63BD = 6\sqrt{3}).
2
Find the lengths of the legs of the isosceles right triangle BCDBCD (45∘−45∘−90∘45^\circ-45^\circ-90^\circ) using the hypotenuse BDBD.
BC=CD=36 cmBC = CD = 3\sqrt{6}\text{ cm}
In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle, the length of each leg is the hypotenuse divided by 2\sqrt{2}. Thus, BC=CD=632=63⋅22=36BC = CD = \frac{6\sqrt{3}}{\sqrt{2}} = \frac{6\sqrt{3}\cdot\sqrt{2}}{2} = 3\sqrt{6}.
3
Calculate the perimeter of quadrilateral ABCDABCD by summing the lengths of its four outer boundary sides: ABAB, BCBC, CDCD, and DADA.
Perimeter =6+36+36+12=18+66 cm= 6 + 3\sqrt{6} + 3\sqrt{6} + 12 = 18 + 6\sqrt{6}\text{ cm}
The perimeter is the sum of the outer boundary sides of the quadrilateral, which are ABAB, BCBC, CDCD, and DADA.

Key Concept

Pythagorean Theorem and Special Right Triangles

Alternative Method

Instead of using special right triangle formulas, the Pythagorean theorem can be used with variables: AB2+BD2=AD2AB^2 + BD^2 = AD^2, where AB=12AD=6AB = \frac{1}{2}AD = 6, so 36+BD2=144  ⟹  BD=108=6336 + BD^2 = 144 \implies BD = \sqrt{108} = 6\sqrt{3}. Then BC2+CD2=BD2  ⟹  2BC2=108  ⟹  BC=54=36BC^2 + CD^2 = BD^2 \implies 2BC^2 = 108 \implies BC = \sqrt{54} = 3\sqrt{6}.
Estimated Time:1m 30s
Question 115Question →

A circular metal plate has a radius of 6 inches6\text{ inches}. A sector of the plate with a central angle of 60∘60^\circ is cut out to make a custom spacer. What is the area, in square inches, of the cut-out sector?

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Answer: 6π6\pi

Answer

The correct area of the sector is 6π6\pi square inches.
The correct answer of 6π6\pi is found by calculating the total area of the circular plate using A=πr2=π(6)2=36πA = \pi r^2 = \pi (6)^2 = 36\pi, and then multiplying this by the ratio of the central angle to the total degrees in a circle, which is 60∘360∘=16\frac{60^\circ}{360^\circ} = \frac{1}{6}. Thus, the sector area is 16×36π=6π\frac{1}{6} \times 36\pi = 6\pi square inches.

Step-by-Step Solution

1
Calculate the area of the entire circular plate.
The total area of the circle is A=π(6)2=36πA = \pi (6)^2 = 36\pi square inches.
Before finding the area of a sector, we need the total area of the circle of which it is a part.
2
Determine the fraction of the circle represented by the sector.
The fraction is 60∘360∘=16\frac{60^\circ}{360^\circ} = \frac{1}{6}.
A circle contains 360∘360^\circ, so a central angle of 60∘60^\circ corresponds to 60360\frac{60}{360} of the full circle.
3
Multiply the total area of the circle by the fraction.
The sector area is 36π×16=6π36\pi \times \frac{1}{6} = 6\pi square inches.
The area of a sector is proportional to its central angle relative to the total angle of a circle.

Key Concept

The area of a sector with radius rr and central angle θ\theta in degrees is given by the formula A=πr2(θ360∘)A = \pi r^2 \left(\frac{\theta}{360^\circ}\right).

Alternative Method

Alternatively, since 60∘60^\circ is 16\frac{1}{6} of a full 360∘360^\circ circle, the sector area is simply one-sixth of the total area of the circle (36π36\pi), which gives 6π6\pi square inches.
Estimated Time:45s
Question 116Question →

In trapezoid ABCDABCD, the bases ABAB and CDCD are parallel. The diagonals ACAC and BDBD intersect at point EE. If the area of △ABE\triangle ABE is 16 cm216\text{ cm}^2 and the area of △CDE\triangle CDE is 36 cm236\text{ cm}^2, what is the total area, in square centimeters, of trapezoid ABCDABCD?

Show answer & explanation

Answer: 100

Answer

The total area of trapezoid ABCDABCD is 100 cm2100\text{ cm}^2.
Triangles ABEABE and CDECDE are similar with an area ratio of 16:3616:36, yielding a side length ratio of 2:32:3. The adjacent triangles △ADE\triangle ADE and △BCE\triangle BCE each share an altitude with the base triangles, making their areas equal to the geometric mean of the base areas: 16×36=24 cm2\sqrt{16 \times 36} = 24\text{ cm}^2 each. Summing the four regions gives 16+36+24+24=100 cm216 + 36 + 24 + 24 = 100\text{ cm}^2.

Step-by-Step Solution

1
Establish the similarity of triangles ABEABE and CDECDE.
△ABE∼△CDE\triangle ABE \sim \triangle CDE
Since bases ABAB and CDCD are parallel, alternate interior angles are congruent (∠EAB≅∠ECD\angle EAB \cong \angle ECD and ∠EBA≅∠EDC\angle EBA \cong \angle EDC). By AA Similarity, the triangles are similar.
2
Calculate the linear scale factor between the similar triangles.
AEEC=BEED=1636=23\frac{AE}{EC} = \frac{BE}{ED} = \sqrt{\frac{16}{36}} = \frac{2}{3}
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths.
3
Find the areas of the adjacent triangles △ADE\triangle ADE and △BCE\triangle BCE.
Area(△ADE)=24 cm2\text{Area}(\triangle ADE) = 24\text{ cm}^2 and Area(△BCE)=24 cm2\text{Area}(\triangle BCE) = 24\text{ cm}^2
Triangles ADEADE and CDECDE share the same altitude from vertex DD to diagonal ACAC, meaning the ratio of their areas is equal to the ratio of their bases: Area(△ADE)Area(△CDE)=AEEC=23  ⟹  Area(△ADE)=23×36=24\frac{\text{Area}(\triangle ADE)}{\text{Area}(\triangle CDE)} = \frac{AE}{EC} = \frac{2}{3} \implies \text{Area}(\triangle ADE) = \frac{2}{3} \times 36 = 24. By similar logic, Area(△BCE)Area(△ABE)=ECAE=32  ⟹  Area(△BCE)=32×16=24\frac{\text{Area}(\triangle BCE)}{\text{Area}(\triangle ABE)} = \frac{EC}{AE} = \frac{3}{2} \implies \text{Area}(\triangle BCE) = \frac{3}{2} \times 16 = 24.
4
Sum the areas of the four individual triangles.
Area(ABCD)=16+36+24+24=100 cm2\text{Area}(ABCD) = 16 + 36 + 24 + 24 = 100\text{ cm}^2
The total area of the trapezoid is the sum of the areas of the four non-overlapping triangles formed by its diagonals.

Key Concept

For any trapezoid with diagonals intersecting at EE and parallel bases forming triangles of areas A1A_1 and A2A_2, the other two triangles each have an area equal to A1A2\sqrt{A_1 A_2}, and the total area of the trapezoid is given by (A1+A2)2(\sqrt{A_1} + \sqrt{A_2})^2.
Question 117Question →

In the figure below, quadrilateral ABCDABCD is composed of two right triangles, △ABC\triangle ABC and △ACD\triangle ACD. The measure of ∠ABC\angle ABC is 90∘90^\circ, and the measure of ∠ACD\angle ACD is 90∘90^\circ. The side lengths are AB=3AB = 3 units and BC=4BC = 4 units. If the measure of ∠CAD\angle CAD is 60∘60^\circ, what is the length, in units, of segment CDCD?

Show answer & explanation

Answer: 535\sqrt{3}

Answer

The length of segment CDCD is 535\sqrt{3} units.
The correct answer is the length of 535\sqrt{3} units. By using the Pythagorean theorem on the first right triangle △ABC\triangle ABC, the length of the hypotenuse is AC=32+42=5AC = \sqrt{3^2 + 4^2} = 5. Since △ACD\triangle ACD is a 30-60-90 right triangle with a right angle at CC and ∠CAD=60∘\angle CAD = 60^\circ, the side ACAC is the shorter leg (opposite the 30∘30^\circ angle). The length of the longer leg CDCD (opposite the 60∘60^\circ angle) is therefore AC3=53AC\sqrt{3} = 5\sqrt{3}.

Step-by-Step Solution

1
Use the Pythagorean theorem in right triangle △ABC\triangle ABC to find the length of the hypotenuse ACAC.
AC=5AC = 5
Since △ABC\triangle ABC is a right triangle with legs AB=3AB = 3 and BC=4BC = 4, the hypotenuse is AC=32+42=9+16=5AC = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5.
2
Identify the type of right triangle for △ACD\triangle ACD.
△ACD\triangle ACD is a 30-60-90 right triangle.
The triangle has a right angle at CC (measure of ∠ACD=90∘\angle ACD = 90^\circ) and an acute angle at AA (measure of ∠CAD=60∘\angle CAD = 60^\circ), which leaves the remaining angle ∠ADC=30∘\angle ADC = 30^\circ.
3
Apply the special right triangle ratios to find the length of leg CDCD.
CD=53CD = 5\sqrt{3}
In a 30-60-90 triangle, the leg opposite the 60∘60^\circ angle is 3\sqrt{3} times the leg opposite the 30∘30^\circ angle. Here, AC=5AC = 5 is opposite the 30∘30^\circ angle, so the longer leg CD=AC3=53CD = AC\sqrt{3} = 5\sqrt{3}.

Key Concept

Using the Pythagorean theorem to find a shared side and then applying special right triangle ratios (30-60-90) to solve for an unknown length.

Alternative Method

Instead of using the special right triangle ratios directly, right triangle trigonometry can be applied: tan⁡(60∘)=oppositeadjacent=CDAC\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{AC}. Since tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3} and AC=5AC = 5, we have 3=CD5\sqrt{3} = \frac{CD}{5}, which yields CD=53CD = 5\sqrt{3}.
Estimated Time:1m 0s
Question 118Question →

In parallelogram ABCDABCD, the ratio of the measure of angle AA to the measure of angle BB is 2:32:3. The measure of angle CC is 1212 degrees less than 33 times the value of xx. What is the value of xx?

Show answer & explanation

Answer: 28

Answer

28
The correct answer is 28. In a parallelogram, consecutive angles are supplementary, so the measure of angle AA and the measure of angle BB must sum to 180∘180^\circ. Based on the given ratio of 2:32:3, the measure of angle AA is calculated as 25×180∘=72∘\frac{2}{5} \times 180^\circ = 72^\circ. Since opposite angles in a parallelogram are equal, the measure of angle CC is also 72∘72^\circ. The description '12 degrees less than 3 times the value of xx' translates to the expression 3x−123x - 12. Equating this to 72∘72^\circ gives 3x−12=723x - 12 = 72, which simplifies to 3x=843x = 84 and yields x=28x = 28.

Step-by-Step Solution

1
Determine the measures of angles AA and BB using their ratio and the properties of a parallelogram.
The measure of angle AA is 72∘72^\circ and the measure of angle BB is 108∘108^\circ.
Adjacent angles in a parallelogram are supplementary, meaning their sum is 180∘180^\circ. Given the ratio of the measure of angle AA to the measure of angle BB is 2:32:3, we can express their measures as 2y2y and 3y3y respectively. Solving 2y+3y=180∘2y + 3y = 180^\circ gives 5y=180∘  ⟹  y=36∘5y = 180^\circ \implies y = 36^\circ. Therefore, the measure of angle AA is 2(36∘)=72∘2(36^\circ) = 72^\circ and the measure of angle BB is 3(36∘)=108∘3(36^\circ) = 108^\circ.
2
Relate the measure of angle CC to the calculated angle measures.
The measure of angle CC is 72∘72^\circ.
Opposite angles of a parallelogram are equal in measure. Since the measure of angle AA is 72∘72^\circ, the measure of the opposite angle CC must also be 72∘72^\circ.
3
Set up and solve the algebraic equation to find the value of xx.
x=28x = 28
We are given that the measure of angle CC is 1212 degrees less than 33 times the value of xx, which translates to 3x−123x - 12. Setting this expression equal to 72∘72^\circ gives the equation 3x−12=723x - 12 = 72. Adding 1212 to both sides yields 3x=843x = 84, and dividing by 33 gives x=28x = 28.

Key Concept

Properties of Parallelograms
Estimated Time:1m 30s
Question 119Question →

A convex hexagon has interior angles with measures of 2x∘2x^\circ, (3x−15)∘(3x - 15)^\circ, (2x+25)∘(2x + 25)^\circ, (3x+10)∘(3x + 10)^\circ, (4x−50)∘(4x - 50)^\circ, and (x+30)∘(x + 30)^\circ. What is the degree measure of the smallest interior angle of this hexagon?

Show answer & explanation

Answer: 78

Answer

The degree measure of the smallest interior angle is 78.
The sum of the interior angles of a hexagon is calculated as (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ. Adding the algebraic expressions for the six angles yields 15x=72015x = 720, which gives x=48x = 48. Substituting x=48x = 48 back into the expressions gives the angle measures of 96∘96^\circ, 129∘129^\circ, 121∘121^\circ, 154∘154^\circ, 142∘142^\circ, and 78∘78^\circ. The smallest angle is 78∘78^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of the hexagon.
Sum of interior angles = 720 degrees
The sum of the interior angles of any convex polygon with nn sides is given by (n−2)×180∘(n - 2) \times 180^\circ. For a hexagon (n=6n = 6), the sum is (6−2)×180∘=720∘(6 - 2) \times 180^\circ = 720^\circ.
2
Set up an algebraic equation by summing all the given angle expressions and equating them to 720.
2x+(3x−15)+(2x+25)+(3x+10)+(4x−50)+(x+30)=7202x + (3x - 15) + (2x + 25) + (3x + 10) + (4x - 50) + (x + 30) = 720
The sum of the actual measures of the interior angles must equal the calculated sum of 720 degrees.
3
Combine like terms to simplify the equation.
15x=72015x = 720
Grouping the xx terms (2x+3x+2x+3x+4x+x=15x2x + 3x + 2x + 3x + 4x + x = 15x) and the constant terms (−15+25+10−50+30=0-15 + 25 + 10 - 50 + 30 = 0) simplifies the expression.
4
Solve for xx.
x=48x = 48
Dividing both sides of the equation 15x=72015x = 720 by 15 isolates the variable xx.
5
Substitute the value of xx back into the angle expressions to identify the smallest angle.
The angles are 96∘96^\circ, 129∘129^\circ, 121∘121^\circ, 154∘154^\circ, 142∘142^\circ, and 78∘78^\circ. The smallest measure is 78∘78^\circ.
Evaluating each expression at x=48x = 48 determines the actual angle measures, from which the smallest can be chosen. Evaluating (x+30)∘(x + 30)^\circ gives 48+30=78∘48 + 30 = 78^\circ, which is the minimum value.

Key Concept

The sum of the interior angles of a convex nn-sided polygon is (n−2)×180∘(n - 2) \times 180^\circ. Setting up and solving linear algebraic equations is required to determine unknown angle measures.
Estimated Time:1m 30s
Question 120Question →

A pendulum of a certain length swings back and forth such that the tip of the pendulum traces an arc of length 6π6\pi inches. If the length of the pendulum is increased by 44 inches and it swings through the same central angle, the tip of the pendulum traces an arc of length 8π8\pi inches. What is the area, in square inches, of the circular sector swept out by the original pendulum?

Show answer & explanation

Answer: 36π36\pi

Answer

The correct answer is 36π36\pi square inches.
The correct answer is 36π36\pi square inches. By using the arc length formula s=rθs = r\theta where θ\theta is in radians, we establish the system of equations rθ=6πr\theta = 6\pi and (r+4)θ=8π(r+4)\theta = 8\pi. Subtracting the first equation from the second gives 4θ=2π4\theta = 2\pi, which simplifies to θ=π2\theta = \frac{\pi}{2}. Substituting this back into the first equation yields r(π2)=6πr(\frac{\pi}{2}) = 6\pi, so the original radius r=12r = 12. Finally, we calculate the area of the original sector using the formula A=12r2θ=12(12)2(π2)=36πA = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2(\frac{\pi}{2}) = 36\pi.

Step-by-Step Solution

1
Write down the arc length equations for both pendulums using the formula s=rθs = r\theta, where rr is the length of the pendulum (radius) and θ\theta is the central angle in radians.
For the original pendulum: rθ=6πr\theta = 6\pi. For the extended pendulum: (r+4)θ=8π(r+4)\theta = 8\pi.
To establish the mathematical relationships between the given arc lengths, the pendulum lengths, and the central angle.
2
Solve the system of equations for the central angle θ\theta.
Distributing the second equation gives rθ+4θ=8πr\theta + 4\theta = 8\pi. Substituting rθ=6πr\theta = 6\pi into this yields 6π+4θ=8π  ⟹  4θ=2π  ⟹  θ=π26\pi + 4\theta = 8\pi \implies 4\theta = 2\pi \implies \theta = \frac{\pi}{2} radians.
To find the constant central angle of the pendulum's swing.
3
Substitute θ=π2\theta = \frac{\pi}{2} back into the first equation to solve for the original radius rr.
r(π2)=6π  ⟹  r=12r\left(\frac{\pi}{2}\right) = 6\pi \implies r = 12 inches.
To find the length of the original pendulum, which serves as the radius of the sector.
4
Calculate the area of the sector swept out by the original pendulum using the formula A=12r2θA = \frac{1}{2}r^2\theta.
A=12(12)2(π2)=12(144)(π2)=36πA = \frac{1}{2}(12)^2\left(\frac{\pi}{2}\right) = \frac{1}{2}(144)\left(\frac{\pi}{2}\right) = 36\pi square inches.
To find the final area of the sector as requested by the question.

Key Concept

Calculating sector area using arc length relationships to determine radius and angle.

Alternative Method

Instead of solving for θ\theta first, one can note that the ratio of the arc lengths is equal to the ratio of the radii because the central angle is constant: rr+4=6π8π=34\frac{r}{r+4} = \frac{6\pi}{8\pi} = \frac{3}{4}. Solving for rr gives 4r=3r+12  ⟹  r=124r = 3r + 12 \implies r = 12. Since the arc length of the original sector is s=6πs = 6\pi, we can use the sector area formula A=12rs=12(12)(6π)=36πA = \frac{1}{2}rs = \frac{1}{2}(12)(6\pi) = 36\pi square inches.
Estimated Time:3m 0s
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