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Plane Geometry

218 questions

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Question 61Question →

In △PQR\triangle PQR, the measure of ∠P\angle P is 50∘50^\circ. The angle bisectors of ∠PQR\angle PQR and ∠PRQ\angle PRQ intersect at point II inside the triangle. What is the measure of ∠QIR\angle QIR?

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Answer: 115∘115^\circ

Answer

115 degrees
To find the measure of the angle at the intersection, we first use the Triangle Angle-Sum Theorem on the larger triangle. The sum of the interior angles in the larger triangle is 180 degrees. Since the angle at one vertex is 50 degrees, the sum of the other two angles must be 130 degrees. The angle bisectors divide these two angles in half, meaning the sum of the two half-angles in the smaller triangle is half of 130 degrees, which is 65 degrees. Applying the Triangle Angle-Sum Theorem to the smaller triangle, the sum of its interior angles is also 180 degrees. Subtracting the sum of the two half-angles (65 degrees) from 180 degrees gives 115 degrees for the angle at the intersection.

Step-by-Step Solution

1
Find the sum of the remaining interior angles of the triangle.
The sum of the angles at Q and R is 130 degrees.
The sum of all interior angles in any triangle is always 180 degrees, and the angle at P is given as 50 degrees.
2
Determine the sum of the bisected angles in the smaller triangle.
The sum of the half-angles is 65 degrees.
Since the lines QI and RI are angle bisectors, the sum of the interior angles of the smaller triangle at vertices Q and R is half the sum of the angles at Q and R of the larger triangle.
3
Calculate the measure of the angle at the intersection point.
The angle at the intersection point is 115 degrees.
The sum of the interior angles in the smaller triangle is also 180 degrees, so the unknown angle is found by subtracting the sum of the two half-angles from 180 degrees.

Key Concept

The Triangle Angle-Sum Theorem states that the sum of the measures of the interior angles of a triangle is always 180 degrees. Angle bisectors divide an angle into two equal parts.

Alternative Method

Alternatively, one can assign specific values to the angles that satisfy the given conditions. Since the sum of the angles in the large triangle must be 180 degrees and the angle at vertex P is 50 degrees, the sum of the other two angles must be 130 degrees. If we assume the triangle is isosceles with the two unknown angles being equal, each of those angles is 65 degrees. Their bisectors would each form an angle of 32.5 degrees with the base. In the smaller triangle, the sum of these two bisected angles is 65 degrees, which leaves 115 degrees for the angle at the intersection point.
Estimated Time:1m 15s
Question 62Question →

In parallelogram ABCDABCD, the measure of interior angle AA is 72∘72^\circ. What is the measure, in degrees, of interior angle BB?

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Answer: 108

Answer

The measure of interior angle BB is 108108 degrees.
In parallelogram ABCDABCD, interior angles AA and BB are consecutive angles. A fundamental property of parallelograms is that consecutive angles are supplementary (their measures sum to 180∘180^\circ). Therefore, the measure of angle BB is calculated as 180∘−72∘=108∘180^\circ - 72^\circ = 108^\circ.

Step-by-Step Solution

1
Use the consecutive angles property of parallelograms.
The sum of consecutive interior angles AA and BB is 180∘180^\circ.
Since opposite sides of a parallelogram are parallel, consecutive interior angles are supplementary.
2
Set up the equation and solve for the unknown angle.
m∠B=108∘m\angle B = 108^\circ
Subtract 72∘72^\circ from 180∘180^\circ.

Key Concept

Properties of parallelograms (consecutive angles are supplementary)
Question 63Question →

In the right trapezoid ABCDABCD below, ABAB is parallel to CDCD, and the measures of ∠A\angle A and ∠D\angle D are both 90∘90^\circ. The length of CDCD is 77, the length of BCBC is 88, and the measure of ∠B\angle B is 60∘60^\circ. What is the length of the diagonal BDBD?

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Answer: 13

Answer

13
By drawing altitude CECE perpendicular to ABAB, we form rectangle AECDAECD and right triangle △CEB\triangle CEB. Since ∠B=60∘\angle B = 60^\circ, △CEB\triangle CEB is a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle with hypotenuse BC=8BC = 8. The side opposite 30∘30^\circ is BE=8/2=4BE = 8/2 = 4, and the side opposite 60∘60^\circ is CE=43CE = 4\sqrt{3}. Since opposite sides of rectangle AECDAECD are equal, we find AD=CE=43AD = CE = 4\sqrt{3} and AE=CD=7AE = CD = 7. Thus, AB=AE+BE=7+4=11AB = AE + BE = 7 + 4 = 11. Finally, we apply the Pythagorean Theorem to right triangle △DAB\triangle DAB: BD2=AD2+AB2=(43)2+112=48+121=169BD^2 = AD^2 + AB^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169, yielding BD=13BD = 13.

Step-by-Step Solution

1
Decompose the trapezoid by drawing an altitude from CC perpendicular to ABAB, meeting it at EE.
This forms a rectangle AECDAECD and a right triangle CEBCEB.
Decomposing the figure allows us to use right triangle trigonometry and parallel line relationships to determine missing side lengths.
2
Calculate the lengths of the legs of right triangle CEBCEB.
BE=4BE = 4 and CE=43CE = 4\sqrt{3}.
Since ∠B=60∘\angle B = 60^\circ, △CEB\triangle CEB is a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle. The shorter leg BEBE is half the hypotenuse BCBC, and the longer leg CECE is the shorter leg times 3\sqrt{3}.
3
Determine the lengths of ADAD and ABAB.
AD=43AD = 4\sqrt{3} and AB=11AB = 11.
In the rectangle AECDAECD, opposite sides are equal, so AD=CE=43AD = CE = 4\sqrt{3} and AE=CD=7AE = CD = 7. Thus, the base AB=AE+BE=7+4=11AB = AE + BE = 7 + 4 = 11.
4
Use the Pythagorean Theorem in right triangle DABDAB to solve for BDBD.
BD=13BD = 13.
In right triangle DABDAB, the legs are AD=43AD = 4\sqrt{3} and AB=11AB = 11. The hypotenuse BDBD satisfies BD2=AD2+AB2=(43)2+112=48+121=169BD^2 = AD^2 + AB^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169, so BD=169=13BD = \sqrt{169} = 13.

Key Concept

Solving multi-step geometry problems by decomposing shapes into rectangles and special right triangles (30∘30^\circ-60∘60^\circ-90∘90^\circ), then applying the Pythagorean Theorem.
Question 64Question →

A convex hexagon has four interior angles that measure 100∘100^\circ, 115∘115^\circ, 125∘125^\circ, and 140∘140^\circ. The remaining two interior angles are in the ratio 3:53:5. What is the degree measure of the largest interior angle of this hexagon?

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Answer: 150∘150^\circ

Answer

The degree measure of the largest interior angle is 150∘150^\circ.
The sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the four given angles (100∘100^\circ, 115∘115^\circ, 125∘125^\circ, and 140∘140^\circ) from 720∘720^\circ leaves a remaining sum of 240∘240^\circ. Since the two remaining angles are in the ratio 3:53:5, they can be written as 3x3x and 5x5x, where 3x+5x=240∘3x + 5x = 240^\circ. Solving for xx gives 8x=240∘  ⟹  x=30∘8x = 240^\circ \implies x = 30^\circ. The two remaining angles are 90∘90^\circ and 150∘150^\circ. Comparing all six interior angles of the hexagon, the largest measure is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a 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-sided polygon is given by (n−2)×180∘(n-2) \times 180^\circ.
2
Find the sum of the four given interior angles.
100∘+115∘+125∘+140∘=480∘100^\circ + 115^\circ + 125^\circ + 140^\circ = 480^\circ.
This determines the total measure of the known angles.
3
Determine the remaining sum of the two unknown angles.
720∘−480∘=240∘720^\circ - 480^\circ = 240^\circ.
Subtracting the sum of the known angles from the total sum yields the combined measure of the remaining two angles.
4
Set up an equation using the ratio 3:53:5 to find the measures of the two remaining angles.
3x+5x=240∘  ⟹  8x=240∘  ⟹  x=30∘3x + 5x = 240^\circ \implies 8x = 240^\circ \implies x = 30^\circ. The two angles are 3(30∘)=90∘3(30^\circ) = 90^\circ and 5(30∘)=150∘5(30^\circ) = 150^\circ.
Representing the two angles in terms of a common variable xx allows us to solve for their individual measures.
5
Identify the largest interior angle among all six angles of the hexagon.
The largest angle is 150∘150^\circ.
Comparing all six angles (90∘90^\circ, 100∘100^\circ, 115∘115^\circ, 125∘125^\circ, 140∘140^\circ, and 150∘150^\circ), the maximum value is 150∘150^\circ.

Key Concept

Calculating the interior angle sum of a polygon and solving for unknown angles using ratios.
Question 65Question →

An irregular convex hexagon has three interior angles that are congruent to each other, and the remaining three interior angles have measures in the ratio 3:4:53:4:5. If the sum of the measures of the three congruent angles is 360∘360^\circ, what is the degree measure of the largest interior angle of the hexagon?

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Answer: 150∘150^\circ

Answer

The degree measure of the largest interior angle of the hexagon is 150∘150^\circ.
The correct answer is 150∘150^\circ. The total sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the sum of the three congruent angles (360∘360^\circ) leaves 360∘360^\circ for the remaining three angles. With their measures in the ratio 3:4:53:4:5, we can write the equation 3x+4x+5x=360∘3x + 4x + 5x = 360^\circ, which simplifies to 12x=360∘12x = 360^\circ, giving x=30∘x = 30^\circ. The measures of these three angles are 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. The three congruent angles each measure 360∘/3=120∘360^\circ / 3 = 120^\circ. Comparing all six angles (120∘,120∘,120∘,90∘,120∘,150∘120^\circ, 120^\circ, 120^\circ, 90^\circ, 120^\circ, 150^\circ), the largest measure is 150∘150^\circ.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a convex hexagon.
720∘720^\circ
Using the formula (n−2)×180∘(n - 2) \times 180^\circ for a polygon with n=6n = 6 sides, the sum is (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Determine the sum of the remaining three interior angles.
360∘360^\circ
We subtract the sum of the three congruent angles (360∘360^\circ) from the total sum of the hexagon's interior angles (720∘720^\circ): 720∘−360∘=360∘720^\circ - 360^\circ = 360^\circ.
3
Set up and solve an equation using the ratio of the remaining three angles.
x=30∘x = 30^\circ
Let the measures of the remaining three angles be 3x3x, 4x4x, and 5x5x. Their sum is 360∘360^\circ, so 3x+4x+5x=360∘  ⟹  12x=360∘  ⟹  x=30∘3x + 4x + 5x = 360^\circ \implies 12x = 360^\circ \implies x = 30^\circ.
4
Calculate the measures of all six interior angles to identify the largest one.
The angles are 120∘120^\circ, 120∘120^\circ, 120∘120^\circ, 90∘90^\circ, 120∘120^\circ, and 150∘150^\circ. The largest is 150∘150^\circ.
Each of the three congruent angles measures 360∘/3=120∘360^\circ / 3 = 120^\circ. The other three angles measure 3(30∘)=90∘3(30^\circ) = 90^\circ, 4(30∘)=120∘4(30^\circ) = 120^\circ, and 5(30∘)=150∘5(30^\circ) = 150^\circ. Comparing all these values, the maximum is 150∘150^\circ.

Key Concept

The sum of the interior angles of a convex nn-sided polygon is given by (n−2)×180∘(n-2) \times 180^\circ. Irregular polygons share this total sum, and ratio relationships can be solved by introducing a variable multiplier.
Question 66Question →

The measures of five of the interior angles of a convex hexagon are in the ratio 3:4:5:6:73:4:5:6:7. The measure of the sixth interior angle is 30∘30^\circ less than the average measure of the other five angles. What is the degree measure of the largest interior angle of the hexagon?

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Answer: 175∘175^\circ

Answer

The correct answer is 175^\circ because solving the equation for the sum of the hexagon's interior angles yields a multiplier of x=25x = 25, making the largest angle 7x=175∘7x = 175^\circ.
The correct answer is 175° because the sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Representing the five ratio-based angles as 3x3x, 4x4x, 5x5x, 6x6x, and 7x7x gives their sum as 25x25x and their average as 5x5x. The sixth angle is therefore 5x−305x - 30. Setting up the sum of all six angles yields 25x+(5x−30)=720  ⟹  30x=750  ⟹  x=2525x + (5x - 30) = 720 \implies 30x = 750 \implies x = 25. The largest angle is 7x=7(25)=175∘7x = 7(25) = 175^\circ.

Step-by-Step Solution

1
Find the sum of the interior angles of a hexagon.
Sum = 720∘720^\circ
Using the interior angle sum formula S=(n−2)×180∘S = (n - 2) \times 180^\circ for a hexagon where n=6n = 6, we get S=(6−2)×180∘=4×180∘=720∘S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
2
Express the five ratio-based angles in terms of a variable xx.
The angles are 3x3x, 4x4x, 5x5x, 6x6x, and 7x7x.
Since the measures of five angles are in the ratio 3:4:5:6:73:4:5:6:7, we can define them as multiples of a common scale factor xx.
3
Compute the average measure of these five angles in terms of xx.
Average = 5x5x
The sum of the five angles is 3x+4x+5x+6x+7x=25x3x + 4x + 5x + 6x + 7x = 25x. The average is the sum divided by the count: 25x5=5x\frac{25x}{5} = 5x.
4
Express the measure of the sixth angle in terms of xx.
Sixth angle = 5x−305x - 30
The sixth angle is described as being 30∘30^\circ less than the average of the other five angles, which we found to be 5x5x.
5
Write and solve the equation for the sum of all six interior angles.
x=25x = 25
The sum of all six angles must equal the total sum of 720∘720^\circ: 25x+(5x−30)=720  ⟹  30x−30=720  ⟹  30x=750  ⟹  x=2525x + (5x - 30) = 720 \implies 30x - 30 = 720 \implies 30x = 750 \implies x = 25.
6
Calculate the measure of the largest interior angle.
175∘175^\circ
The largest interior angle corresponds to the largest term in the ratio, which is 7x7x. Substituting x=25x = 25 yields 7×25=175∘7 \times 25 = 175^\circ.

Key Concept

Applying the interior angle sum formula for polygons combined with ratio and algebraic translation properties.

Alternative Method

Instead of using the sum of interior angles, one could use the sum of exterior angles, which is always 360∘360^\circ. The exterior angles corresponding to the five angles in ratio 3:4:5:6:73:4:5:6:7 would be 180−3x180 - 3x, 180−4x180 - 4x, 180−5x180 - 5x, 180−6x180 - 6x, and 180−7x180 - 7x. The sixth exterior angle is 180−(5x−30)=210−5x180 - (5x - 30) = 210 - 5x. Summing these six exterior angles: (180−3x)+(180−4x)+(180−5x)+(180−6x)+(180−7x)+(210−5x)=1110−30x=360(180 - 3x) + (180 - 4x) + (180 - 5x) + (180 - 6x) + (180 - 7x) + (210 - 5x) = 1110 - 30x = 360. Solving for xx yields 30x=750  ⟹  x=2530x = 750 \implies x = 25. The largest interior angle corresponds to the smallest exterior angle, which is 180−7x=180−7(25)=5∘180 - 7x = 180 - 7(25) = 5^\circ, giving the largest interior angle as 180−5=175∘180 - 5 = 175^\circ.
Estimated Time:3m 0s
Question 67Question →

In rectangle PQRSPQRS, the length of side PQPQ is 8 inches and the length of diagonal PRPR is 10 inches. What is the length, in inches, of side QRQR?

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Answer: 6

Answer

The length of side QR is 6 inches.
The correct answer is 6. In a rectangle, all four interior angles are right angles (90 degrees). Therefore, triangle PQR is a right triangle with the right angle at Q. According to the Pythagorean theorem, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: PQ² + QR² = PR². Substituting the given values: 8² + QR² = 10², which simplifies to 64 + QR² = 100. Subtracting 64 from both sides gives QR² = 36. Taking the square root of both sides yields QR = 6 inches.

Step-by-Step Solution

1
Identify the geometric relationship in rectangle PQRS.
Angle PQR is a right angle (90 degrees), which makes triangle PQR a right triangle with legs PQ and QR, and hypotenuse PR.
By definition, all interior angles of a rectangle are 90 degrees, and the diagonal connects opposite vertices, forming two right triangles.
2
Apply the Pythagorean theorem to triangle PQR.
PQ² + QR² = PR², which becomes 8² + QR² = 10².
The Pythagorean theorem states that in any right triangle, the sum of the squares of the leg lengths equals the square of the hypotenuse length.
3
Solve for the unknown side length QR.
64 + QR² = 100, which gives QR² = 36, and thus QR = 6.
Subtract 64 from both sides to isolate QR², then take the square root of 36.

Key Concept

Properties of rectangles and the application of the Pythagorean theorem to right triangles formed by diagonals.
Question 68Question →

In △ABC\triangle ABC, the measure of exterior angle ∠ACD\angle ACD is 135∘135^\circ, where DD lies on the extension of side BCBC past CC. If the measure of interior angle ∠A\angle A is 25∘25^\circ greater than the measure of interior angle ∠B\angle B, what is the measure, in degrees, of ∠B\angle B?

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Answer: 55

Answer

55
According to the Exterior Angle Theorem, the measure of exterior angle ∠ACD\angle ACD is equal to the sum of the measures of its remote interior angles, ∠A\angle A and ∠B\angle B. This gives the equation m∠A+m∠B=135∘\text{m}\angle A + \text{m}\angle B = 135^\circ. Using the information that m∠A=m∠B+25∘\text{m}\angle A = \text{m}\angle B + 25^\circ, we substitute this expression into the equation to get (m∠B+25∘)+m∠B=135∘(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ. Simplifying this equation gives 2m∠B+25=1352\text{m}\angle B + 25 = 135, which simplifies to 2m∠B=1102\text{m}\angle B = 110, and dividing by 2 yields m∠B=55∘\text{m}\angle B = 55^\circ.

Step-by-Step Solution

1
Apply the Exterior Angle Theorem to express the relation between the exterior angle and the two remote interior angles.
m∠A+m∠B=135∘\text{m}\angle A + \text{m}\angle B = 135^\circ
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2
Substitute the relationship between the interior angles into the equation.
(m∠B+25∘)+m∠B=135∘(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ
The problem states that the measure of interior angle ∠A\angle A is 25∘25^\circ greater than the measure of interior angle ∠B\angle B.
3
Solve the algebraic equation for the measure of interior angle ∠B\angle B.
m∠B=55∘\text{m}\angle B = 55^\circ
Combining like terms gives 2m∠B+25=1352\text{m}\angle B + 25 = 135. Subtracting 25 from both sides gives 2m∠B=1102\text{m}\angle B = 110. Dividing by 2 yields m∠B=55∘\text{m}\angle B = 55^\circ.

Key Concept

Exterior Angle Theorem and remote interior angles relation
Question 69Question →

A convex polygon has nn sides. The sum of the measures of its interior angles is 33 times the sum of the measures of its exterior angles. What is the value of nn?

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Answer: 8

Answer

8
The sum of the interior angle measures of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ. The sum of the exterior angle measures is always 360∘360^\circ. According to the problem, the sum of the interior angles is 33 times the sum of the exterior angles, which can be written as the equation (n−2)×180=3×360(n-2) \times 180 = 3 \times 360. Simplifying the right side gives (n−2)×180=1080(n-2) \times 180 = 1080. Dividing both sides by 180180 results in n−2=6n - 2 = 6. Adding 22 to both sides yields n=8n = 8.

Step-by-Step Solution

1
Use the formula for the sum of the interior angle measures of a convex polygon with nn sides.
The sum of the interior angles is (n−2)×180∘(n-2) \times 180^\circ.
By the polygon interior angle sum theorem, the sum of the interior angles of any convex nn-gon is (n−2)×180∘(n-2) \times 180^\circ.
2
Identify the sum of the exterior angle measures of a convex polygon.
The sum of the exterior angles is 360∘360^\circ.
The sum of the exterior angles of any convex polygon is always constant and equals 360∘360^\circ, regardless of the number of sides.
3
Set up the equation relating the two sums as given in the problem statement.
(n−2)×180∘=3×360∘(n-2) \times 180^\circ = 3 \times 360^\circ
The problem states that the sum of the interior angles is 33 times the sum of the exterior angles.
4
Solve the equation for nn.
n=8n = 8
Divide both sides by 180∘180^\circ to get n−2=6n - 2 = 6, then add 22 to both sides to find n=8n = 8.

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 the sum of its exterior angles is 360∘360^\circ.
Question 70Question →

The sum of the measures of the interior angles of a convex polygon is 900∘900^\circ. How many sides does this polygon have?

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Answer: 7

Answer

The polygon has 7 sides.
The correct answer is 77 because the sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n - 2) \times 180^\circ. Setting this equal to 900∘900^\circ yields (n−2)×180=900(n - 2) \times 180 = 900. Dividing both sides by 180180 gives n−2=5n - 2 = 5, and adding 22 to both sides results in n=7n = 7.

Step-by-Step Solution

1
Set up the equation for the sum of the interior angles of a polygon.
(n−2)×180∘=900∘(n - 2) \times 180^\circ = 900^\circ
The sum of the interior angles of a convex polygon with nn sides is always (n−2)×180∘(n - 2) \times 180^\circ.
2
Divide both sides of the equation by 180∘180^\circ.
n−2=5n - 2 = 5
To isolate the term with nn, we divide 900900 by 180180.
3
Solve for nn by adding 22 to both sides.
n=7n = 7
Adding 22 to both sides isolates the variable nn.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n - 2) \times 180^\circ.
Estimated Time:45s
Question 71Question →

A triangle has two sides of length 99 and 1414. The length of the third side, ss, is a multiple of 44. What is the sum of all possible integer values for ss?

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Answer: 56

Answer

56
According to the Triangle Inequality Theorem, the length of the third side, ss, of a triangle with sides of length 9 and 14 must satisfy the inequality 14−9<s<14+914 - 9 < s < 14 + 9, which simplifies to 5<s<235 < s < 23. The integer values within this range that are multiples of 4 are 8, 12, 16, and 20. Adding these values together yields a sum of 56.

Step-by-Step Solution

1
Apply the Triangle Inequality Theorem to determine the range of possible lengths for the third side.
The third side length, ss, must satisfy the inequality: 14−9<s<14+914 - 9 < s < 14 + 9, which simplifies to 5<s<235 < s < 23.
The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the positive difference of the other two sides and strictly less than the sum of the other two sides.
2
Identify all integer values in the range (5,23)(5, 23) that are multiples of 4.
The multiples of 4 that are strictly greater than 5 and strictly less than 23 are: 8, 12, 16, and 20.
We must find the integers within the bounds that can be divided by 4 with a remainder of 0.
3
Calculate the sum of the identified multiples of 4.
8+12+16+20=568 + 12 + 16 + 20 = 56.
The question asks for the sum of all possible integer values for the third side length ss.

Key Concept

Triangle Inequality Theorem
Estimated Time:1m 0s
Question 72Question →

A regular decagon has 1010 sides of equal length. What is the measure, in degrees, of one exterior angle of this decagon?

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Answer: 36

Answer

The correct answer is 36.
The sum of the exterior angles of any convex polygon is always 360∘360^\circ. A regular decagon has 1010 congruent sides and therefore 1010 congruent exterior angles. Dividing 360∘360^\circ by 1010 yields 36∘36^\circ for each exterior angle.

Step-by-Step Solution

1
Determine the number of exterior angles in a regular decagon.
A decagon has 10 sides, so it has 10 exterior angles.
A polygon has the same number of exterior angles as its number of sides.
2
State the sum of the exterior angles for a convex polygon.
The sum of the exterior angles is 360 degrees.
The exterior angles of any convex polygon always sum to 360 degrees regardless of the number of sides.
3
Calculate the measure of one exterior angle.
360 / 10 = 36
Since the decagon is regular, all of its exterior angles are equal in measure, so we divide the total sum by the number of angles.

Key Concept

The sum of the exterior angles of any convex polygon is 360∘360^\circ. For a regular polygon with nn sides, the measure of each exterior angle is 360∘n\frac{360^\circ}{n}.
Question 73Question →

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

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Answer: 60∘60^\circ

Answer

60∘60^\circ
The sum of the interior angles of a convex pentagon is calculated as (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. Representing the five angle measures in the given ratio as 2x2x, 3x3x, 4x4x, 4x4x, and 5x5x, their sum is 18x=540∘18x = 540^\circ, which gives x=30∘x = 30^\circ. The smallest angle corresponds to the smallest term in the ratio, which is 2x=2×30∘=60∘2x = 2 \times 30^\circ = 60^\circ. Therefore, the correct measure is 60∘60^\circ.

Step-by-Step Solution

1
Determine the sum of the interior angles of a convex pentagon.
The sum is (5−2)×180∘=3×180∘=540∘(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
The sum of the interior angles of any convex nn-gon is given by the formula (n−2)×180∘(n - 2) \times 180^\circ, and a pentagon has 55 sides.
2
Set up an algebraic equation using the given ratio of the angle measures.
Let the angle measures be 2x2x, 3x3x, 4x4x, 4x4x, and 5x5x. Their sum is 2x+3x+4x+4x+5x=18x=540∘2x + 3x + 4x + 4x + 5x = 18x = 540^\circ.
The sum of the actual angle measures must equal the total interior angle sum of the pentagon.
3
Solve for the variable xx.
x=540∘/18=30∘x = 540^\circ / 18 = 30^\circ.
Dividing the total sum by the sum of the ratio parts gives the value of a single ratio unit.
4
Calculate the measure of the smallest interior angle.
Smallest angle =2x=2×30∘=60∘= 2x = 2 \times 30^\circ = 60^\circ.
The smallest term in the ratio is 22, so multiplying this term by xx gives the smallest angle 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 with angles in a given ratio, the individual angle measures can be found by setting up a linear equation where the sum of the ratio parts multiplied by a variable equals the total sum.
Question 74Question →

Is the following statement true or false? In any rectangle, the diagonals must be perpendicular to each other.

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Answer: False

Answer

False
The correct answer is False because the diagonals of a rectangle are perpendicular only when the rectangle has equal side lengths (making it a square). In a standard rectangle, the diagonals are congruent and bisect each other, but they do not intersect at right angles.

Step-by-Step Solution

1
Analyze the properties of a general rectangle.
A rectangle is a parallelogram with four right angles. Its diagonals are congruent and bisect each other.
To determine diagonal behavior, we must start with the standard properties of the shape.
2
Examine if the diagonals are perpendicular in all cases.
The diagonals of a rectangle are perpendicular only if the rectangle is a square (all sides equal). For a general rectangle with unequal adjacent sides, the diagonals are not perpendicular.
A single counterexample, such as a rectangle with unequal adjacent sides, shows that the property does not hold universally.

Key Concept

Properties of rectangle diagonals
Estimated Time:45s
Question 75Question →

In the standard (x,y)(x, y) coordinate plane, a circle is centered at the origin (0,0)(0,0) and has a radius of 88. A horizontal chord ABAB lies entirely in the first and second quadrants at a distance of 44 units from the xx-axis. A point PP is located on the circle such that △ABP\triangle ABP is a right triangle. If the hypotenuse of △ABP\triangle ABP is a diameter of the circle, what is the area of △ABP\triangle ABP?

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Answer: 32332\sqrt{3}

Answer

The area of the right triangle is 32332\sqrt{3}.
The correct answer is 32332\sqrt{3}. The horizontal chord ABAB has yy-coordinate 44, and its endpoints lie on the circle x2+y2=64x^2 + y^2 = 64. Solving for xx gives x=±43x = \pm 4\sqrt{3}, so the length of the chord is 838\sqrt{3}. Because the triangle is inscribed in the circle and is a right triangle, its hypotenuse must be a diameter of the circle (length 1616). Since AB<16AB < 16, ABAB is a leg, and the hypotenuse is one of the other sides (e.g., APAP). The remaining leg BPBP is found using the Pythagorean theorem: BP=162−(83)2=256−192=64=8BP = \sqrt{16^2 - (8\sqrt{3})^2} = \sqrt{256 - 192} = \sqrt{64} = 8. The area of the right triangle is 12×83×8=323\frac{1}{2} \times 8\sqrt{3} \times 8 = 32\sqrt{3}.

Step-by-Step Solution

1
Determine the length of chord ABAB.
The length of chord ABAB is 838\sqrt{3}.
The equation of the circle is x2+y2=64x^2 + y^2 = 64. Since the chord is horizontal and at a distance of 44 units from the xx-axis, its yy-coordinate is 44. Substituting y=4y = 4 gives x2+16=64⇒x2=48⇒x=±43x^2 + 16 = 64 \Rightarrow x^2 = 48 \Rightarrow x = \pm 4\sqrt{3}. The distance between A(−43,4)A(-4\sqrt{3}, 4) and B(43,4)B(4\sqrt{3}, 4) is 838\sqrt{3}.
2
Apply the rule for a right triangle inscribed in a circle to identify the hypotenuse.
The hypotenuse must be a diameter of length 1616, so the right angle is at BB (or AA).
Any right triangle inscribed in a circle must have a diameter as its hypotenuse. The diameter of this circle is 2×8=162 \times 8 = 16. Since the chord AB=83≈13.86AB = 8\sqrt{3} \approx 13.86 is shorter than the diameter, it cannot be the hypotenuse. Therefore, either APAP or BPBP is the hypotenuse (a diameter), making the angle opposite to it (either ∠ABP\angle ABP or ∠BAP\angle BAP) the 90∘90^\circ angle.
3
Calculate the length of the remaining leg of the right triangle.
The length of leg BPBP is 88.
Using the Pythagorean theorem for right triangle ABPABP with hypotenuse AP=16AP = 16 and leg AB=83AB = 8\sqrt{3}: AB2+BP2=AP2⇒(83)2+BP2=162⇒192+BP2=256⇒BP2=64⇒BP=8AB^2 + BP^2 = AP^2 \Rightarrow (8\sqrt{3})^2 + BP^2 = 16^2 \Rightarrow 192 + BP^2 = 256 \Rightarrow BP^2 = 64 \Rightarrow BP = 8.
4
Compute the area of right triangle ABPABP.
The area is 32332\sqrt{3}.
The area of a right triangle is 12×base×height=12×AB×BP=12×83×8=323\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times BP = \frac{1}{2} \times 8\sqrt{3} \times 8 = 32\sqrt{3}.

Key Concept

Applying the Pythagorean theorem and Thales's theorem (inscribed right triangles) to solve multi-step geometric problems on the coordinate plane.
Question 76Question →

A kite WXYZWXYZ has diagonals WYWY and XZXZ that intersect at point PP. If the length of WPWP is 44 centimeters, the length of PYPY is 99 centimeters, and the length of XPXP is 33 centimeters, what is the measure, in degrees, of angle WPXWPX?

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Answer: 90

Answer

90 degrees
The diagonals of a kite are always perpendicular to each other. Therefore, the angle formed at their intersection, angle WPXWPX, is a right angle, which measures exactly 9090 degrees. The given segment lengths are extra information.

Step-by-Step Solution

1
Identify the fundamental property of the diagonals of a kite.
The diagonals of any kite are perpendicular to each other.
By geometric definition, the diagonals of a kite intersect at a right angle.
2
Determine the measure of the angle formed by the intersection of the diagonals.
Angle WPXWPX is a right angle, which measures exactly 9090 degrees.
Since the diagonals are perpendicular, their intersection forms four 9090-degree angles regardless of the lengths of the individual diagonal segments.

Key Concept

The diagonals of a kite are perpendicular (90∘90^\circ).
Question 77Question →

An irregular convex polygon has nn sides. The measures of its interior angles, in degrees, are all distinct integers. If all of the interior angles are obtuse, what is the maximum possible value of nn?

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Answer: 26

Answer

The maximum possible value of nn is 26.
For a convex 2626-gon, the sum of the interior angles is 24×180∘=4320∘24 \times 180^\circ = 4320^\circ. We can choose 2626 distinct integer angles from the range [91∘,179∘][91^\circ, 179^\circ] that sum to exactly 4320∘4320^\circ because the maximum possible sum of 2626 distinct integers in this range is 4329∘4329^\circ, which is greater than 4320∘4320^\circ. For n=27n = 27, the sum of the interior angles must be 25×180∘=4500∘25 \times 180^\circ = 4500^\circ, but the maximum possible sum of 2727 distinct integers in the range is only 4482∘4482^\circ, which is less than 4500∘4500^\circ. Therefore, 2626 is the maximum value of nn.

Step-by-Step Solution

1
Determine the set of possible angle measures.
The angles must be integers in the range [91∘,179∘][91^\circ, 179^\circ].
Interior angles of a convex polygon must be less than 180∘180^\circ. Since they are obtuse and distinct integers, they must be strictly greater than 90∘90^\circ, giving the range [91,179][91, 179].
2
Write the sum of the interior angles of a convex nn-gon.
Sum =(n−2)×180∘= (n - 2) \times 180^\circ.
This is the standard formula for the sum of the interior angles of any convex nn-gon.
3
Find the maximum possible sum of nn distinct angles in the range [91,179][91, 179].
Maximum Sum =179n−n(n−1)2= 179n - \frac{n(n - 1)}{2}.
The maximum sum is achieved by selecting the largest nn integers from the set: 179,178,…,179−(n−1)179, 178, \dots, 179 - (n - 1).
4
Set up the inequality and simplify.
n2+n−720≤0n^2 + n - 720 \le 0.
Since the sum of the angles must be less than or equal to the maximum possible sum, we have (n−2)×180≤179n−n(n−1)2(n-2) \times 180 \le 179n - \frac{n(n-1)}{2}. Multiplying by 2 and simplifying yields the quadratic inequality.
5
Solve the quadratic inequality for the largest integer nn.
n=26n = 26.
Evaluating the quadratic expression for consecutive integers: for n=26n = 26, 262+26−720=−18≤026^2 + 26 - 720 = -18 \le 0; for n=27n = 27, 272+27−720=36>027^2 + 27 - 720 = 36 > 0. Thus, 26 is the maximum possible integer value.

Key Concept

Sum of interior angles of a convex polygon combined with algebraic optimization.
Question 78Question →

If the diagonals of a convex quadrilateral divide the quadrilateral into four triangles of equal perimeter, then the quadrilateral must be a square. Is this statement true or false?

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Answer: False

Answer

The statement is false because any non-square rhombus satisfies the condition of being divided into four triangles of equal perimeter, yet it is not a square.
The correct answer is false because the equal-perimeter condition only requires the quadrilateral to be a rhombus (having four equal sides), but does not require the interior angles to be 90 degrees. Any non-square rhombus is a valid counterexample.

Step-by-Step Solution

1
Analyze the given condition of equal perimeters for the four triangles formed by the diagonals of a convex quadrilateral.
The diagonals of any rhombus are perpendicular and bisect each other, dividing the rhombus into four congruent right triangles.
Congruent triangles have identical side lengths, meaning their perimeters are equal. Thus, every rhombus satisfies this property.
2
Determine if all quadrilaterals satisfying this property must be squares.
A square is a regular quadrilateral, meaning it must have both equal side lengths and interior angles of 90 degrees.
To verify if the statement is true, we must test if a non-square rhombus can satisfy the condition.
3
Construct a counterexample using a specific non-square rhombus.
Consider a rhombus with side lengths of 5 units and diagonals of lengths 6 units and 8 units. The diagonals divide it into four right triangles with sides 3, 4, and 5 units. Each triangle has a perimeter of 12 units.
This rhombus has equal perimeters for all four triangles, but its interior angles are not 90 degrees, proving it is not a square.

Key Concept

The relationship between the diagonals and side properties of rhombuses and squares.
Question 79Question →

A convex polygon has nn sides. The sum of the measures of its interior angles is 66 times the sum of the measures of its exterior angles (one at each vertex). What is the value of nn?

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Answer: 14

Answer

The number of sides, nn, of the convex polygon is 14.
The sum of the interior angles of a convex polygon with nn sides is given by the formula (n−2)×180∘(n-2) \times 180^\circ, and the sum of its exterior angles is always 360∘360^\circ. According to the problem, the sum of the interior angles is 66 times the sum of the exterior angles, yielding the equation (n−2)×180=6×360(n - 2) \times 180 = 6 \times 360. Dividing both sides of the equation by 180180 gives n−2=12n - 2 = 12. Adding 22 to both sides results in n=14n = 14.

Step-by-Step Solution

1
State the sum of interior and exterior angles formulas.
Interior sum = (n−2)×180∘(n-2) \times 180^\circ, Exterior sum = 360∘360^\circ.
To represent the geometric properties of the polygon algebraically.
2
Set up the equation based on the given ratio.
(n−2)×180=6×360(n-2) \times 180 = 6 \times 360.
The problem states the interior sum is 6 times the exterior sum.
3
Solve the equation for nn.
n=14n = 14.
Divide by 180 to get n−2=12n - 2 = 12, then add 2 to both sides.

Key Concept

The sum of the interior angles of an nn-sided convex polygon is (n−2)×180∘(n-2) \times 180^\circ, and the sum of the exterior angles (one per vertex) is always 360∘360^\circ.
Question 80Question →

An irregular convex hexagon has two interior angles measuring 90∘90^\circ and 130∘130^\circ, respectively. The remaining four interior angles have measures in the ratio 5:6:7:75:6:7:7. What is the degree measure of the largest interior angle in this hexagon?

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Answer: 140∘140^\circ

Answer

The correct answer is 140∘140^\circ.
The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the two given angles (90∘90^\circ and 130∘130^\circ) leaves 500∘500^\circ for the remaining four angles. Since these angles are in the ratio 5:6:7:75:6:7:7, we represent them as 5x5x, 6x6x, 7x7x, and 7x7x. Their sum is 25x=500∘25x = 500^\circ, which gives x=20∘x = 20^\circ. The largest of the remaining angles is 7x=7(20∘)=140∘7x = 7(20^\circ) = 140^\circ. Since 140∘140^\circ is greater than both 90∘90^\circ and 130∘130^\circ, it is the largest interior angle of the hexagon.

Step-by-Step Solution

1
Calculate the sum of the interior angles of a hexagon.
Using the formula for the sum of the interior angles of a polygon with nn sides, (n−2)×180∘(n - 2) \times 180^\circ, for a hexagon (n=6n = 6), the sum is (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
To establish the total sum of all interior angles of the polygon.
2
Subtract the two known angle measures from the total sum.
720∘−90∘−130∘=500∘720^\circ - 90^\circ - 130^\circ = 500^\circ.
To find the sum of the remaining four interior angles.
3
Set up an algebraic equation to find the value of one ratio unit, xx.
5x+6x+7x+7x=500∘  ⟹  25x=500∘  ⟹  x=20∘5x + 6x + 7x + 7x = 500^\circ \implies 25x = 500^\circ \implies x = 20^\circ.
To determine the constant multiplier for the ratio of the remaining angles.
4
Calculate the measures of the remaining angles and determine the largest angle.
The remaining angles are 5(20∘)=100∘5(20^\circ) = 100^\circ, 6(20∘)=120∘6(20^\circ) = 120^\circ, 7(20∘)=140∘7(20^\circ) = 140^\circ, and 7(20∘)=140∘7(20^\circ) = 140^\circ. Comparing all six angles of the hexagon (90∘,100∘,120∘,130∘,140∘,140∘90^\circ, 100^\circ, 120^\circ, 130^\circ, 140^\circ, 140^\circ), the largest angle is 140∘140^\circ.
To identify the maximum interior angle measure of the hexagon.

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n - 2) \times 180^\circ. The individual angle measures in an irregular polygon can be determined using algebraic representations of their relationships or ratios.

Alternative Method

Once the value of the ratio unit x=20∘x = 20^\circ is determined, you can quickly find the largest candidate angle by multiplying the largest ratio component (77) by xx to get 7(20∘)=140∘7(20^\circ) = 140^\circ, and then compare it to the given angles (90∘90^\circ and 130∘130^\circ) to verify that it is indeed the largest.
Estimated Time:1m 30s
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