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Question 2541Question →

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 2542Question →

In the standard (x,y)(x, y) coordinate system, a line given by the equation y=2x+4y = 2x + 4 intersects a parabola given by the equation y=x2−2x−1y = x^2 - 2x - 1 at exactly two points. What is the sum of the yy-coordinates of these two points of intersection?

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Answer: 16

Answer

16
The system of equations is solved by setting the equations equal to each other, yielding the quadratic equation x2−4x−5=0x^2 - 4x - 5 = 0. Solving for xx gives x=5x = 5 and x=−1x = -1. Substituting these values into the linear equation gives the y-coordinates 1414 and 22. The sum of these y-coordinates is 14+2=1614 + 2 = 16.

Step-by-Step Solution

1
Equate the linear and quadratic equations to find the x-values of the intersection points.
x2−2x−1=2x+4x^2 - 2x - 1 = 2x + 4
At the points of intersection, the y-values of both equations must be equal.
2
Rearrange the equation to standard quadratic form.
x2−4x−5=0x^2 - 4x - 5 = 0
Grouping all terms on one side allows the quadratic equation to be solved.
3
Factor the quadratic equation to find the x-coordinates.
(x−5)(x+1)=0(x - 5)(x + 1) = 0, so x=5x = 5 or x=−1x = -1
The numbers that multiply to −5-5 and add up to −4-4 are −5-5 and 11.
4
Substitute the x-coordinates into the linear equation to find the y-coordinates.
For x=5x = 5: y=2(5)+4=14y = 2(5) + 4 = 14. For x=−1x = -1: y=2(−1)+4=2y = 2(-1) + 4 = 2.
Evaluating the linear equation is simpler than evaluating the quadratic equation.
5
Find the sum of the y-coordinates.
14+2=1614 + 2 = 16
The question asks for the sum of the y-coordinates of the two points of intersection.

Key Concept

Solving a system of linear and quadratic equations by substitution.

Alternative Method

Instead of solving for the individual intersection points, Vieta's formulas can be applied. The x-coordinates satisfy x2−4x−5=0x^2 - 4x - 5 = 0, so their sum is x1+x2=4x_1 + x_2 = 4. Since the points lie on the line y=2x+4y = 2x + 4, the sum of the y-coordinates is y1+y2=(2x1+4)+(2x2+4)=2(x1+x2)+8=2(4)+8=16y_1 + y_2 = (2x_1 + 4) + (2x_2 + 4) = 2(x_1 + x_2) + 8 = 2(4) + 8 = 16.
Estimated Time:1m 30s
Question 2543Question →

A parallelogram with vertices P(−3,−2)P(-3, -2), Q(5,−2)Q(5, -2), R(8,2)R(8, 2), and S(0,2)S(0, 2) is plotted on a standard coordinate grid. What is the perimeter of parallelogram PQRSPQRS?

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Answer: 26

Answer

The correct answer is 26, which is the perimeter of the parallelogram.
The horizontal base segments have a length of 5−(−3)=85 - (-3) = 8. The slanted side has a length of (0−(−3))2+(2−(−2))2=32+42=5\sqrt{(0 - (-3))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = 5. Summing the four sides gives a perimeter of 2(8)+2(5)=262(8) + 2(5) = 26.

Step-by-Step Solution

1
Calculate the length of the horizontal base PQ using the x-coordinates of P(-3, -2) and Q(5, -2).
PQ=5−(−3)=8PQ = 5 - (-3) = 8
Horizontal segments on the coordinate plane have a length equal to the difference in their x-coordinates.
2
Apply the distance formula to find the length of the slanted side PS with vertices P(-3, -2) and S(0, 2).
PS=(0−(−3))2+(2−(−2))2=32+42=9+16=5PS = \sqrt{(0 - (-3))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
The distance formula determines the straight-line distance between two coordinates.
3
Calculate the total perimeter of parallelogram PQRS by summing the lengths of all four sides.
Perimeter=2×base+2×slanted side=2(8)+2(5)=16+10=26\text{Perimeter} = 2 \times \text{base} + 2 \times \text{slanted side} = 2(8) + 2(5) = 16 + 10 = 26
The perimeter of a parallelogram is the sum of all its outer boundary lengths.

Key Concept

Calculating the perimeter of geometric figures on the coordinate plane by determining horizontal distances and applying the distance formula for skewed lines.

Alternative Method

Instead of using the distance formula, you can sketch a right triangle for the slanted side PS. The horizontal leg has length 3 and the vertical leg has length 4. Recognizing this as a standard 3-4-5 right triangle, the hypotenuse (the slanted side) must be 5.
Estimated Time:1m 30s
Question 2544Question →

On a map of a city, Oak Street is represented by a straight line with the equation 3x−4y=123x - 4y = 12. A new road, Pine Street, is planned to be perpendicular to Oak Street and will pass through a park located at the coordinates (2,−1)(2, -1). Which of the following is an equation that represents Pine Street?

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Answer: 4x+3y=54x + 3y = 5

Answer

4x+3y=54x + 3y = 5
The slope of Oak Street is 34\frac{3}{4}. The perpendicular line representing Pine Street must have a slope that is the negative reciprocal of 34\frac{3}{4}, which is −43-\frac{4}{3}. Using the point-slope form with the point (2,−1)(2, -1), we get y−(−1)=−43(x−2)y - (-1) = -\frac{4}{3}(x - 2). Simplifying this yields y+1=−43x+83y + 1 = -\frac{4}{3}x + \frac{8}{3}, which becomes y=−43x+53y = -\frac{4}{3}x + \frac{5}{3}. Multiplying the entire equation by 3 and moving the xx term to the left side gives the standard form equation 4x+3y=54x + 3y = 5.

Step-by-Step Solution

1
Find the slope of Oak Street by converting its equation to slope-intercept form (y=mx+by = mx + b).
The equation 3x−4y=123x - 4y = 12 becomes −4y=−3x+12-4y = -3x + 12, which simplifies to y=34x−3y = \frac{3}{4}x - 3. The slope of Oak Street is 34\frac{3}{4}.
To find the slope of a perpendicular line, we must first determine the slope of the original line.
2
Determine the perpendicular slope of Pine Street by taking the negative reciprocal of Oak Street's slope.
The negative reciprocal of 34\frac{3}{4} is −43-\frac{4}{3}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Use the point-slope formula y−y1=m(x−x1)y - y_1 = m(x - x_1) with the point (2,−1)(2, -1) and the perpendicular slope −43-\frac{4}{3} to find the equation of Pine Street.
Substituting the values gives y−(−1)=−43(x−2)y - (-1) = -\frac{4}{3}(x - 2), which simplifies to y+1=−43x+83y + 1 = -\frac{4}{3}x + \frac{8}{3}.
The point-slope formula allows us to write the equation of a line given its slope and a point it passes through.
4
Convert the equation to standard form (Ax+By=CAx + By = C).
Subtract 1 from both sides: y=−43x+83−33⇒y=−43x+53y = -\frac{4}{3}x + \frac{8}{3} - \frac{3}{3} \Rightarrow y = -\frac{4}{3}x + \frac{5}{3}. Multiply the entire equation by 3: 3y=−4x+53y = -4x + 5. Add 4x4x to both sides: 4x+3y=54x + 3y = 5.
Converting to standard form matches the format of the options provided in the question.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (m1⋅m2=−1m_1 \cdot m_2 = -1). Once the perpendicular slope is found, the point-slope formula can be used to write the equation of the line passing through a given point.
Question 2545Question →

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 2546Question →

A segment in the coordinate plane has one endpoint at A(1,9)A(1, 9) and its midpoint at M(5,6)M(5, 6). Let BB represent the other endpoint of the segment. What is the distance between point BB and the point C(3,−5)C(3, -5)?

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Answer: 1010

Answer

The distance between point BB and point CC is 1010.
First, we find the coordinates of point B(x,y)B(x, y) using the midpoint formula. Since M(5,6)M(5, 6) is the midpoint between A(1,9)A(1, 9) and B(x,y)B(x, y), we set up the equations: 5=1+x25 = \frac{1+x}{2} which simplifies to 10=1+x10 = 1+x and x=9x = 9; and 6=9+y26 = \frac{9+y}{2} which simplifies to 12=9+y12 = 9+y and y=3y = 3. Thus, point BB is (9,3)(9, 3). Next, we apply the distance formula to find the distance between B(9,3)B(9, 3) and C(3,−5)C(3, -5): d=(9−3)2+(3−(−5))2=62+82=36+64=100=10d = \sqrt{(9-3)^2 + (3-(-5))^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10. This is the correct distance.

Step-by-Step Solution

1
Find the coordinates of endpoint B(xB,yB)B(x_B, y_B) using the midpoint formula.
xB=9x_B = 9 and yB=3y_B = 3, so point BB is (9,3)(9, 3).
Since M(5,6)M(5, 6) is the midpoint of segment ABAB, we can solve for BB's coordinates using 5=1+xB25 = \frac{1 + x_B}{2} and 6=9+yB26 = \frac{9 + y_B}{2}.
2
Calculate the distance between B(9,3)B(9, 3) and C(3,−5)C(3, -5) using the distance formula.
d=10d = 10
Applying the distance formula: d=(9−3)2+(3−(−5))2=62+82=100=10d = \sqrt{(9-3)^2 + (3-(-5))^2} = \sqrt{6^2 + 8^2} = \sqrt{100} = 10.

Key Concept

Using the midpoint formula to find a missing endpoint, and then applying the distance formula between two coordinate points.
Question 2547Question →

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 2548Question →

A model glider is launched from a hill. Its height h(t)h(t), in meters above the valley floor tt seconds after launch, is modeled by the function h(t)=−0.5t2+3.5t+10h(t) = -0.5t^2 + 3.5t + 10. Which of the following is a possible value of tt, in seconds, when the glider is at a height of exactly 1212 meters?

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Answer: 7+332\frac{7 + \sqrt{33}}{2}

Answer

7+332\frac{7 + \sqrt{33}}{2}
The correct answer is obtained by setting the height equation h(t)=12h(t) = 12, which simplifies to −0.5t2+3.5t−2=0-0.5t^2 + 3.5t - 2 = 0. Multiplying by −2-2 gives the standard form t2−7t+4=0t^2 - 7t + 4 = 0. Applying the quadratic formula with a=1a = 1, b=−7b = -7, and c=4c = 4 gives t=7±332t = \frac{7 \pm \sqrt{33}}{2}. Thus, the option representing 7+332\frac{7 + \sqrt{33}}{2} is the correct choice.

Step-by-Step Solution

1
Set the height function equal to the target height of 12 meters.
−0.5t2+3.5t+10=12-0.5t^2 + 3.5t + 10 = 12
To find when the glider reaches exactly 12 meters, we set the model function equal to 12.
2
Subtract 12 from both sides to set the quadratic equation to zero.
−0.5t2+3.5t−2=0-0.5t^2 + 3.5t - 2 = 0
A quadratic equation must be in standard form at2+bt+c=0at^2 + bt + c = 0 before applying the quadratic formula.
3
Multiply the entire equation by −2-2 to eliminate decimal coefficients.
t2−7t+4=0t^2 - 7t + 4 = 0
Working with integer coefficients reduces calculation errors when applying the quadratic formula. Here, a=1a = 1, b=−7b = -7, and c=4c = 4.
4
Apply the quadratic formula t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
t=−(−7)±(−7)2−4(1)(4)2(1)=7±49−162=7±332t = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(4)}}{2(1)} = \frac{7 \pm \sqrt{49 - 16}}{2} = \frac{7 \pm \sqrt{33}}{2}
The quadratic formula is used to solve quadratic equations that cannot be easily factored using integers.

Key Concept

Solving quadratic equations with decimal coefficients by converting to standard integer form and applying the quadratic formula.
Question 2549Question →

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 2550Question →

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 2551Question →

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 2552Question →

For all real values of nn that satisfy the inequality 3−2n4<3n−225\frac{3 - 2n}{4} < \frac{3n - 22}{5}, what is the smallest possible integer value of nn?

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Answer: 5

Answer

The smallest integer value of nn that satisfies the inequality is 5.
Solving the inequality yields n>10322≈4.68n > \frac{103}{22} \approx 4.68. The smallest integer greater than 4.684.68 is 55.

Step-by-Step Solution

1
Multiply both sides by 20 to clear the denominators.
5(3−2n)<4(3n−22)5(3 - 2n) < 4(3n - 22)
To eliminate the fractions and simplify the inequality.
2
Distribute the constants on both sides.
15−10n<12n−8815 - 10n < 12n - 88
To remove the parentheses.
3
Subtract 12n12n from both sides.
15−22n<−8815 - 22n < -88
To group the terms containing the variable on the left side.
4
Subtract 15 from both sides.
−22n<−103-22n < -103
To isolate the variable term on the left side.
5
Divide both sides by −22-22 and reverse the inequality sign.
n>10322n > \frac{103}{22}
Dividing by a negative number reverses the direction of the inequality sign.
6
Convert the fraction to a decimal to identify the boundary.
n>4.68n > 4.68
To find the smallest integer value that satisfies this condition.
7
Identify the smallest integer greater than 4.684.68.
55
The smallest integer greater than 4.684.68 is 55.

Key Concept

Solving linear inequalities by applying inverse operations and reversing the inequality sign when multiplying or dividing by a negative number.
Estimated Time:1m 30s
Question 2553Question →

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 2554Question →

In the standard (x,y)(x, y) coordinate plane, square PQRSPQRS has adjacent vertices at P(1,2)P(1, 2) and Q(4,6)Q(4, 6). The line containing the side QRQR has a y-intercept of bb. What is the value of bb?

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Answer: 9

Answer

The correct answer is 9.
The slope of segment PQPQ is 6−24−1=43\frac{6 - 2}{4 - 1} = \frac{4}{3}. Since adjacent sides of a square are perpendicular, the line containing QRQR is perpendicular to PQPQ and passes through Q(4,6)Q(4, 6). The slope of this perpendicular line is the negative reciprocal of 43\frac{4}{3}, which is −34-\frac{3}{4}. Using the slope-intercept form y=mx+by = mx + b with point Q(4,6)Q(4, 6), we substitute the values: 6=−34(4)+b⇒6=−3+b⇒b=96 = -\frac{3}{4}(4) + b \Rightarrow 6 = -3 + b \Rightarrow b = 9.

Step-by-Step Solution

1
Calculate the slope of side PQPQ using the coordinates of P(1,2)P(1, 2) and Q(4,6)Q(4, 6).
The slope of PQPQ is mPQ=6−24−1=43m_{PQ} = \frac{6 - 2}{4 - 1} = \frac{4}{3}.
To determine the direction of side PQPQ so we can find the perpendicular slope for QRQR.
2
Find the slope of the line containing side QRQR.
The slope of QRQR is mQR=−34m_{QR} = -\frac{3}{4}.
Because adjacent sides of a square are perpendicular, the slope of QRQR is the negative reciprocal of the slope of PQPQ.
3
Find the equation of the line containing QRQR using the slope-intercept form and the coordinates of vertex Q(4,6)Q(4, 6).
Substituting the slope m=−34m = -\frac{3}{4} and point (4,6)(4, 6) into y=mx+by = mx + b gives 6=−34(4)+b6 = -\frac{3}{4}(4) + b, which simplifies to 6=−3+b6 = -3 + b, so b=9b = 9.
The line containing side QRQR must pass through vertex QQ, which allows us to determine the y-intercept bb.

Key Concept

The slopes of perpendicular lines are negative reciprocals of each other: m1⋅m2=−1m_1 \cdot m_2 = -1.
Question 2555Question →

In the standard (x,y)(x, y) coordinate plane, a trapezoid has vertices at A(−5,−4)A(-5, -4), B(7,−4)B(7, -4), C(7,1)C(7, 1), and D(−5,6)D(-5, 6). What is the perimeter of this trapezoid?

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Answer: 40

Answer

The perimeter of the trapezoid is 40.
The perimeter of the trapezoid is found by summing the lengths of its four sides. Two sides are vertical (AD of length 10 and BC of length 5), one side is horizontal (AB of length 12), and the final side is skewed (CD of length 13, found using the distance formula). Summing these gives 10 + 5 + 12 + 13 = 40.

Step-by-Step Solution

1
Calculate the lengths of the vertical sides AD and BC
AD = 10, BC = 5
Since vertices A(-5, -4) and D(-5, 6) share the same x-coordinate of -5, the side AD is vertical. Its length is the difference in their y-coordinates: 6 - (-4) = 10. Similarly, vertices B(7, -4) and C(7, 1) share the same x-coordinate of 7, so the side BC is vertical. Its length is the difference in their y-coordinates: 1 - (-4) = 5.
2
Calculate the length of the horizontal side AB
AB = 12
Since vertices A(-5, -4) and B(7, -4) share the same y-coordinate of -4, the side AB is horizontal. Its length is the difference in their x-coordinates: 7 - (-5) = 12.
3
Calculate the length of the skewed side CD using the distance formula
CD = 13
Using the coordinates of C(7, 1) and D(-5, 6), the distance is calculated as: CD = \sqrt{(-5 - 7)^2 + (6 - 1)^2} = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.
4
Sum the lengths of all sides to find the perimeter
Perimeter = 40
The perimeter of a polygon is the sum of the lengths of all its sides: Perimeter = AB + BC + CD + DA = 12 + 5 + 13 + 10 = 40.

Key Concept

Calculating side lengths of a geometric figure on a coordinate plane to find its perimeter
Question 2556Question →

Which of the following inequalities represents the complete solution set for xx in the inequality x3−2x−52≥2\frac{x}{3} - \frac{2x - 5}{2} \ge 2?

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Answer: x≤34x \le \frac{3}{4}

Answer

The inequality is satisfied by all values of xx less than or equal to 34\frac{3}{4}.
The correct solution is obtained by rewriting the fractions with a common denominator of 6, which yields 2x−3(2x−5)6≥2\frac{2x - 3(2x - 5)}{6} \ge 2. Distributing the negative sign gives −4x+156≥2\frac{-4x + 15}{6} \ge 2. Multiplying both sides by 6 and subtracting 15 results in −4x≥−3-4x \ge -3. Dividing by −4-4 and reversing the inequality sign yields x≤34x \le \frac{3}{4}.

Step-by-Step Solution

1
Find a common denominator of 6 for the two fractions on the left side of the inequality.
2x6−3(2x−5)6≥2\frac{2x}{6} - \frac{3(2x - 5)}{6} \ge 2
Finding a common denominator is necessary to combine fractional terms with different denominators.
2
Combine the fractions into a single expression, distributing the subtraction sign to both terms in the numerator of the second fraction.
2x−(6x−15)6≥2  ⟹  −4x+156≥2\frac{2x - (6x - 15)}{6} \ge 2 \implies \frac{-4x + 15}{6} \ge 2
Combining the fractions simplifies the inequality. Distributing the negative sign ensures correct operations on the terms.
3
Multiply both sides of the inequality by 6.
−4x+15≥12-4x + 15 \ge 12
Multiplying by a positive number isolates the numerator without changing the direction of the inequality sign.
4
Subtract 15 from both sides of the inequality.
−4x≥−3-4x \ge -3
This isolates the variable term on the left side.
5
Divide both sides by −4-4 and reverse the direction of the inequality sign.
x≤34x \le \frac{3}{4}
Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain the correct relationship.

Key Concept

Solving linear inequalities requires applying standard algebraic operations while remembering to reverse the inequality sign when multiplying or dividing both sides by a negative number.

Alternative Method

Instead of combining fractions first, multiply every term on both sides of the inequality by 6 (the least common multiple of the denominators) to eliminate the fractions immediately: 6⋅(x3)−6⋅(2x−52)≥6⋅26 \cdot \left(\frac{x}{3}\right) - 6 \cdot \left(\frac{2x - 5}{2}\right) \ge 6 \cdot 2. This simplifies directly to 2x−3(2x−5)≥122x - 3(2x - 5) \ge 12, which reduces to −4x+15≥12-4x + 15 \ge 12.
Estimated Time:1m 30s
Question 2557Question →

On a coordinate map of an airport, Runway A is represented by the line 4x−3y=124x - 3y = 12. Runway B is designed to be perpendicular to Runway A, and its path crosses the yy-axis at (0,7)(0, 7). If Runway B passes through a guidance beacon located at the coordinate (p,1)(p, 1), what is the value of pp?

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Answer: 88

Answer

8
The correct answer is 88. First, find the slope of Runway A by rewriting 4x−3y=124x - 3y = 12 as y=43x−4y = \frac{4}{3}x - 4, which gives a slope of 43\frac{4}{3}. The slope of Runway B must be the negative reciprocal, −34-\frac{3}{4}. Using the given yy-intercept of 77, the equation for Runway B is y=−34x+7y = -\frac{3}{4}x + 7. Substituting the point (p,1)(p, 1) gives 1=−34p+71 = -\frac{3}{4}p + 7. Subtracting 77 from both sides yields −6=−34p-6 = -\frac{3}{4}p. Multiplying both sides by −43-\frac{4}{3} gives p=8p = 8.

Step-by-Step Solution

1
Convert the equation of Runway A, 4x−3y=124x - 3y = 12, into slope-intercept form (y=mx+by = mx + b) to find its slope.
The equation becomes y=43x−4y = \frac{4}{3}x - 4, which shows the slope of Runway A is m1=43m_1 = \frac{4}{3}.
Finding the slope of the first line is necessary to determine the slope of any perpendicular line.
2
Calculate the perpendicular slope (m2m_2) for Runway B by taking the negative reciprocal of the slope of Runway A (m1m_1).
The perpendicular slope is m2=−1m1=−34m_2 = -\frac{1}{m_1} = -\frac{3}{4}.
Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other.
3
Write the equation of Runway B using its slope m2=−34m_2 = -\frac{3}{4} and its yy-intercept of 77.
The equation of Runway B is y=−34x+7y = -\frac{3}{4}x + 7.
The slope-intercept form of a line is y=mx+by = mx + b, where mm is the slope and bb is the yy-intercept.
4
Substitute the point (p,1)(p, 1) into the equation of Runway B and solve for pp.
1=−34p+7  ⟹  −6=−34p  ⟹  24=3p  ⟹  p=81 = -\frac{3}{4}p + 7 \implies -6 = -\frac{3}{4}p \implies 24 = 3p \implies p = 8.
Since the guidance beacon lies on Runway B, its coordinates must satisfy the equation of the line.

Key Concept

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is −1-1. Once the perpendicular slope and y-intercept are known, the line's equation can be written and solved for a missing coordinate.
Estimated Time:1m 0s
Question 2558Question →

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 2559Question →

For what value of cc does the quadratic equation 0.5x2−3x+c=00.5x^2 - 3x + c = 0 have two real solutions that differ by exactly 4?

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Answer: 2.5

Answer

2.5
The correct value is 2.5. By utilizing the formula for the difference of the roots, b2−4ac∣a∣=4\frac{\sqrt{b^2 - 4ac}}{|a|} = 4, and substituting a=0.5a = 0.5 and b=−3b = -3, we get 9−2c0.5=4\frac{\sqrt{9 - 2c}}{0.5} = 4. This simplifies to 9−2c=2\sqrt{9 - 2c} = 2, which squares to 9−2c=49 - 2c = 4. Solving for cc yields 2.5.

Step-by-Step Solution

1
Identify the coefficients and apply the relationship for the difference between two roots.
For a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the roots x1x_1 and x2x_2 satisfy ∣x1−x2∣=b2−4ac∣a∣|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}. Here, a=0.5a = 0.5, b=−3b = -3, and the difference is 4.
This formula relates the difference of the roots directly to the coefficients of the quadratic equation.
2
Substitute the given values into the formula and solve for cc.
Substituting the values gives (−3)2−4(0.5)c∣0.5∣=4⇒9−2c0.5=4\frac{\sqrt{(-3)^2 - 4(0.5)c}}{|0.5|} = 4 \Rightarrow \frac{\sqrt{9 - 2c}}{0.5} = 4. Multiplying both sides by 0.5 yields 9−2c=2\sqrt{9 - 2c} = 2. Squaring both sides gives 9−2c=49 - 2c = 4.
Simplifying the equation isolates the variable cc under the radical.
3
Complete the algebraic isolation to find the final value of cc.
2c=5⇒c=2.52c = 5 \Rightarrow c = 2.5.
This final step solves the linear equation for cc.

Key Concept

Using the discriminant and properties of roots to solve quadratic equations with given constraints.
Question 2560Question →

In the standard (x,y)(x, y) coordinate plane, the line L1L_1 is parallel to the line y=3x−7y = 3x - 7. The line L2L_2 is perpendicular to L1L_1 and passes through the point (6,5)(6, 5). If the yy-intercept of L2L_2 is (0,c)(0, c), what is the value of cc?

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Answer: 7

Answer

The value of cc is 77.
Since the line L1L_1 is parallel to y=3x−7y = 3x - 7, its slope is 33. The line L2L_2 is perpendicular to L1L_1, so its slope is the negative reciprocal of 33, which is −13-\frac{1}{3}. Using the point-slope formula with the point (6,5)(6, 5) yields the equation of L2L_2: y−5=−13(x−6)y - 5 = -\frac{1}{3}(x - 6), which simplifies to y=−13x+7y = -\frac{1}{3}x + 7. The yy-intercept of this line is (0,7)(0, 7), which gives c=7c = 7.

Step-by-Step Solution

1
Determine the slope of line L1L_1.
Slope of L1L_1 is 33.
Parallel lines have equal slopes, and the given reference line y=3x−7y = 3x - 7 has a slope of 33.
2
Determine the slope of line L2L_2.
Slope of L2L_2 is −13-\frac{1}{3}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Find the equation of line L2L_2.
The equation is y=−13x+7y = -\frac{1}{3}x + 7.
Use the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (6,5)(6, 5) and slope −13-\frac{1}{3}.
4
Identify the yy-intercept constant cc.
c=7c = 7.
The equation of L2L_2 is in slope-intercept form y=mx+by = mx + b, meaning the yy-intercept is (0,7)(0, 7).

Key Concept

The relationship between the slopes of parallel lines (which are equal) and perpendicular lines (which are negative reciprocals).
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