All practice questions

The table below shows coordinate points $(x, y)$ that lie on a line $L$ in the standard $(x, y)$ coordinate system:

$x$	$y$
$-2$	$-7$
$1$	$2$
$4$	$11$

What is the $x$ -intercept of line $L$ ?

\frac{1}{3}

-\frac{1}{3}

-1

-5

\frac{5}{3}

Show answer & explanation

Answer: $\frac{1}{3}$

Answer

The correct answer is

\frac{1}{3}

The correct answer is

\frac{1}{3}

. First, find the slope of the line using the points

(-2, -7)

and

(1, 2)

from the table:

m = \frac{2 - (-7)}{1 - (-2)} = 3

. Next, substitute the slope and the coordinate point

(1, 2)

into the point-slope equation to get

y - 2 = 3(x - 1)

, which simplifies to

y = 3x - 1

. To find the

x

-intercept, set

y = 0

and solve for

x

:

0 = 3x - 1

, which gives

x = \frac{1}{3}

.

Step-by-Step Solution

1

Calculate the slope (

m

) of line

L

using the coordinate points

(-2, -7)

and

(1, 2)

from the table.

The slope is

m = \frac{2 - (-7)}{1 - (-2)} = \frac{9}{3} = 3

.

The slope of a line represents its constant rate of change and is calculated using the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Write the linear equation in slope-intercept form using the slope

m = 3

and the point

(1, 2)

.

Using the point-slope form:

y - 2 = 3(x - 1) \Rightarrow y - 2 = 3x - 3 \Rightarrow y = 3x - 1

.

Using point-slope form

y - y_1 = m(x - x_1)

allows us to easily simplify the relationship into slope-intercept form

y = mx + b

.

3

Find the

x

-intercept of the line by setting

y = 0

and solving the equation for

x

.

Setting

y = 0

gives

0 = 3x - 1 \Rightarrow 3x = 1 \Rightarrow x = \frac{1}{3}

.

The

x

-intercept is the coordinate value where the graph intersects the

x

-axis, which mathematically occurs when

y = 0

.

Key Concept

Determining the equation of a line from tabular coordinates and calculating its intercepts

Question 2462Question →

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 $3:4:5$ . If the sum of the measures of the three congruent angles is $360^\circ$ , what is the degree measure of the largest interior angle of the hexagon?

90^\circ

120^\circ

150^\circ

250^\circ

300^\circ

Show answer & explanation

Answer: $150^\circ$

Answer

The degree measure of the largest interior angle of the hexagon is

150^\circ

.

The correct answer is

150^\circ

. The total sum of the interior angles of a hexagon is

(6-2) \times 180^\circ = 720^\circ

. Subtracting the sum of the three congruent angles (

360^\circ

) leaves

360^\circ

for the remaining three angles. With their measures in the ratio

3:4:5

, we can write the equation

3x + 4x + 5x = 360^\circ

, which simplifies to

12x = 360^\circ

, giving

x = 30^\circ

. The measures of these three angles are

90^\circ

,

120^\circ

, and

150^\circ

. The three congruent angles each measure

360^\circ / 3 = 120^\circ

. Comparing all six angles (

120^\circ, 120^\circ, 120^\circ, 90^\circ, 120^\circ, 150^\circ

), the largest measure is

150^\circ

.

Step-by-Step Solution

1

Calculate the sum of the interior angles of a convex hexagon.

720^\circ

Using the formula

(n - 2) \times 180^\circ

for a polygon with

n = 6

sides, the sum is

(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.

2

Determine the sum of the remaining three interior angles.

360^\circ

We subtract the sum of the three congruent angles (

360^\circ

) from the total sum of the hexagon's interior angles (

720^\circ

):

720^\circ - 360^\circ = 360^\circ

.

3

Set up and solve an equation using the ratio of the remaining three angles.

x = 30^\circ

Let the measures of the remaining three angles be

3x

,

4x

, and

5x

. Their sum is

360^\circ

, so

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^\circ

,

120^\circ

,

120^\circ

,

90^\circ

,

120^\circ

, and

150^\circ

. The largest is

150^\circ

.

Each of the three congruent angles measures

360^\circ / 3 = 120^\circ

. The other three angles measure

3(30^\circ) = 90^\circ

,

4(30^\circ) = 120^\circ

, and

5(30^\circ) = 150^\circ

. Comparing all these values, the maximum is

150^\circ

.

Key Concept

The sum of the interior angles of a convex

n

-sided polygon is given by

(n-2) \times 180^\circ

. Irregular polygons share this total sum, and ratio relationships can be solved by introducing a variable multiplier.

Question 2463Question →

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

125^\circ

161^\circ

175^\circ

202^\circ

203^\circ

Show answer & explanation

Answer: $175^\circ$

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 = 25

, making the largest angle

7x = 175^\circ

.

The correct answer is 175° because the sum of the interior angles of a hexagon is

(6-2) \times 180^\circ = 720^\circ

. Representing the five ratio-based angles as

3x

,

4x

,

5x

,

6x

, and

7x

gives their sum as

25x

and their average as

5x

. The sixth angle is therefore

5x - 30

. Setting up the sum of all six angles yields

25x + (5x - 30) = 720 \implies 30x = 750 \implies x = 25

. The largest angle is

7x = 7(25) = 175^\circ

.

Step-by-Step Solution

1

Find the sum of the interior angles of a hexagon.

Sum =

720^\circ

Using the interior angle sum formula

S = (n - 2) \times 180^\circ

for a hexagon where

n = 6

, we get

S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.

2

Express the five ratio-based angles in terms of a variable

x

.

The angles are

3x

,

4x

,

5x

,

6x

, and

7x

.

Since the measures of five angles are in the ratio

3:4:5:6:7

, we can define them as multiples of a common scale factor

x

.

3

Compute the average measure of these five angles in terms of

x

.

Average =

5x

The sum of the five angles is

3x + 4x + 5x + 6x + 7x = 25x

. The average is the sum divided by the count:

\frac{25x}{5} = 5x

.

4

Express the measure of the sixth angle in terms of

x

.

Sixth angle =

5x - 30

The sixth angle is described as being

30^\circ

less than the average of the other five angles, which we found to be

5x

.

5

Write and solve the equation for the sum of all six interior angles.

x = 25

The sum of all six angles must equal the total sum of

720^\circ

:

25x + (5x - 30) = 720 \implies 30x - 30 = 720 \implies 30x = 750 \implies x = 25

.

6

Calculate the measure of the largest interior angle.

175^\circ

The largest interior angle corresponds to the largest term in the ratio, which is

7x

. Substituting

x = 25

yields

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^\circ

. The exterior angles corresponding to the five angles in ratio

3:4:5:6:7

would be

180 - 3x

,

180 - 4x

,

180 - 5x

,

180 - 6x

, and

180 - 7x

. The sixth exterior angle is

180 - (5x - 30) = 210 - 5x

. Summing these six exterior angles:

(180 - 3x) + (180 - 4x) + (180 - 5x) + (180 - 6x) + (180 - 7x) + (210 - 5x) = 1110 - 30x = 360

. Solving for

x

yields

30x = 750 \implies x = 25

. The largest interior angle corresponds to the smallest exterior angle, which is

180 - 7x = 180 - 7(25) = 5^\circ

, giving the largest interior angle as

180 - 5 = 175^\circ

.

Estimated Time:3m 0s

Question 2464Question →

A student translates a word problem into an inequality. The problem states: "One-fourth of the difference of $x$ and $3$ is greater than the sum of $1$ and one-third of the quantity $2x$ minus $1$ ." Which of the following inequalities represents the correct set of all real values of $x$ that satisfy this condition?

x > -\frac{17}{5}

x < -\frac{44}{5}

x < -3

x < -\frac{17}{5}

x < -\frac{8}{3}

Show answer & explanation

Answer: $x < -\frac{17}{5}$

Answer

The set of all real values of

x

satisfying the condition is

x < -\frac{17}{5}

.

The correct inequality is obtained by translating the word problem statement as

\frac{x - 3}{4} > 1 + \frac{2x - 1}{3}

. Multiplying the entire inequality by

12

yields

3(x - 3) > 12 + 4(2x - 1)

. Distributing the coefficients results in

3x - 9 > 12 + 8x - 4

, which simplifies to

3x - 9 > 8x + 8

. Moving all terms containing

x

to the left and constants to the right gives

-5x > 17

. Dividing both sides by

-5

and reversing the inequality sign results in the solution

x < -\frac{17}{5}

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic inequality.

\frac{x - 3}{4} > 1 + \frac{2x - 1}{3}

The phrase "one-fourth of the difference of

x

and

3

" translates to

\frac{x - 3}{4}

, and "the sum of

1

and one-third of the quantity

2x

minus

1

" translates to

1 + \frac{2x - 1}{3}

.

2

Multiply all terms by the least common multiple of the denominators, which is

12

, to clear the fractions.

3(x - 3) > 12 + 4(2x - 1)

Multiplying both sides of an inequality by a positive number maintains the direction of the inequality sign while eliminating fractional coefficients.

3

Expand both sides of the inequality by distributing the coefficients.

3x - 9 > 12 + 8x - 4

Distribution allows the variable and constant terms to be separated and simplified.

4

Simplify the constants on the right side and move all variable terms to one side and constants to the other.

-5x > 17

Subtracting

8x

from both sides and adding

9

to both sides isolates the variable term on the left side of the inequality.

5

Divide both sides by

-5

and reverse the inequality sign.

x < -\frac{17}{5}

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

Key Concept

Solving linear inequalities by translating word problems and applying the sign-flip rule when dividing by a negative number.

Estimated Time:1m 30s

Question 2465Question →

A circle in the standard $(x, y)$ coordinate plane is described by the equation $x^2 + y^2 = 25$ . A line is described by the equation $3x - 4y = c$ , where $c$ is a positive constant. If the system of these two equations has exactly one real solution for $(x, y)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 25

Answer

The value of

c

is

25

.

The correct answer is

25

. The equation

x^2 + y^2 = 25

represents a circle centered at

(0, 0)

with a radius of

5

. For the system of equations to have exactly one real solution, the line

3x - 4y = c

must be tangent to the circle. The perpendicular distance from the center

(0,0)

to the line

3x - 4y - c = 0

is given by

\frac{|3(0) - 4(0) - c|}{\sqrt{3^2 + (-4)^2}} = \frac{|c|}{5}

. Setting this distance equal to the radius of the circle yields

\frac{|c|}{5} = 5

, which gives

|c| = 25

. Since

c

is specified as a positive constant,

c

must be

25

.

Step-by-Step Solution

1

Find the center and radius of the circle.

Center is

(0, 0)

and radius is

r = 5

.

The circle equation

x^2 + y^2 = 25

is in the standard form

x^2 + y^2 = r^2

centered at the origin with radius

r = \sqrt{25} = 5

.

2

Set up the condition for tangency (exactly one real solution).

The perpendicular distance from the center

(0,0)

to the line

3x - 4y - c = 0

must equal the radius

5

.

A line intersects a circle at exactly one point if and only if the line is tangent to the circle.

3

Apply the point-to-line distance formula.

Distance

d = \frac{|3(0) - 4(0) - c|}{\sqrt{3^2 + (-4)^2}} = \frac{|c|}{5}

.

The formula for the distance from

(x_0, y_0)

to the line

Ax + By + C = 0

is

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

.

4

Solve for the positive constant

c

.

c = 25

Setting the distance

\frac{|c|}{5}

equal to the radius

5

gives

|c| = 25

. Since

c

is a positive constant,

c = 25

.

Key Concept

Determining conditions for tangency in a system of linear and circular equations.

Alternative Method

Instead of using the geometric distance formula, the system can be solved algebraically by substitution. Express

y

in terms of

x

from the linear equation:

y = \frac{3x - c}{4}

. Substitute this expression into the circle's equation:

x^2 + \left(\frac{3x - c}{4}\right)^2 = 25

. Expand the terms and multiply by

16

to clear the denominator:

16x^2 + 9x^2 - 6cx + c^2 = 400

, which simplifies to the quadratic equation

25x^2 - 6cx + (c^2 - 400) = 0

. For the system to have exactly one solution, this quadratic equation must have a discriminant equal to zero. Calculate the discriminant:

D = (-6c)^2 - 4(25)(c^2 - 400) = 36c^2 - 100c^2 + 40000 = -64c^2 + 40000 = 0

. Solving for

c

yields

64c^2 = 40000 \implies c^2 = 625

. Since

c

is a positive constant,

c = 25

.

Estimated Time:1m 30s

Question 2466Question →

The first term of an arithmetic sequence is $\frac{1}{2}$ , and the second term is $\frac{5}{6}$ . The first term of a geometric sequence is $2^2$ , and the common ratio is $2^3$ . Let $A$ be the third term of the arithmetic sequence, and let $G$ be the third term of the geometric sequence. If a third value, $V$ , is defined as $\frac{5}{6}$ less than twice $G$ , what is the value of $V + A$ ?

\frac{1541}{3}

512

\frac{1537}{3}

-510

\frac{24577}{3}

Show answer & explanation

Answer: $\frac{1537}{3}$

Answer

\frac{1537}{3}

The correct answer is

\frac{1537}{3}

because calculating the third term of the arithmetic sequence yields

A = \frac{7}{6}

and the third term of the geometric sequence yields

G = 256

. Translating the relationship for

V

yields

V = 2(256) - \frac{5}{6} = 512 - \frac{5}{6}

. Finding the sum of

V

and

A

yields

512 - \frac{5}{6} + \frac{7}{6} = 512\frac{1}{3} = \frac{1537}{3}

.

Step-by-Step Solution

1

Calculate the common difference

d

of the arithmetic sequence.

d = a_2 - a_1 = \frac{5}{6} - \frac{1}{2} = \frac{1}{3}

The common difference is the difference between any term and the preceding term in an arithmetic sequence.

2

Calculate the third term

A

of the arithmetic sequence.

A = a_1 + 2d = \frac{1}{2} + 2\left(\frac{1}{3}\right) = \frac{7}{6}

The

n

-th term of an arithmetic sequence is given by

a_n = a_1 + (n-1)d

.

3

Calculate the third term

G

of the geometric sequence.

G = g_1 \cdot r^2 = 2^2 \cdot (2^3)^2 = 2^2 \cdot 2^6 = 2^8 = 256

The

n

-th term of a geometric sequence is given by

g_n = g_1 \cdot r^{n-1}

.

4

Set up and solve for

V

using the algebraic relationship described.

V = 2G - \frac{5}{6} = 2(256) - \frac{5}{6} = 512 - \frac{5}{6}

The phrase '5/6 less than twice G' translates to

2G - \frac{5}{6}

.

5

Compute the sum of

V

and

A

.

V + A = \left(512 - \frac{5}{6}\right) + \frac{7}{6} = 512 + \frac{2}{6} = 512 + \frac{1}{3} = \frac{1537}{3}

Substitute the values of

V

and

A

and simplify the resulting fractional expression.

Key Concept

Arithmetic and Geometric Sequences and Series

Estimated Time:1m 30s

Question 2467Question →

One of the solutions to the quadratic equation $0.5x^2 + bx - 6 = 0$ , where $b$ is a constant, is $x = 3$ . What is the value of the other solution?

Show answer & explanation

Answer: -4

Answer

The other solution to the quadratic equation is -4.

Substituting the given solution

x = 3

into the equation yields

0.5(3)^2 + 3b - 6 = 0

. Simplifying this expression gives

4.5 + 3b - 6 = 0

, which leads to

3b = 1.5

and thus

b = 0.5

. With

b = 0.5

, the quadratic equation becomes

0.5x^2 + 0.5x - 6 = 0

. Multiplying the entire equation by 2 to obtain integer coefficients results in

x^2 + x - 12 = 0

. This quadratic factors into

(x - 3)(x + 4) = 0

, which gives the solutions

x = 3

and

x = -4

. Therefore, the other solution is

-4

. Alternatively, using Vieta's formulas, the product of the roots of a quadratic equation

ax^2 + bx + c = 0

is equal to

c/a

. Here, the product of the roots is

-6 / 0.5 = -12

. Since one root is

3

, the other root must be

-12 / 3 = -4

.

Step-by-Step Solution

1

Substitute the given solution

x = 3

into the quadratic equation to find the value of

b

.

b = 0.5

Since

x = 3

is a solution, it must satisfy the equation, allowing us to solve for the unknown coefficient

b

.

2

Rewrite the equation using

b = 0.5

and simplify by multiplying all terms by 2.

x^2 + x - 12 = 0

Multiplying the equation by 2 eliminates the decimal coefficients, making the quadratic expression easier to factor.

3

Factor the quadratic equation to determine the roots.

x = 3

or

x = -4

The equation factors into

(x - 3)(x + 4) = 0

. Solving for

x

yields the given root of 3 and the second root of -4.

Key Concept

Solving quadratic equations by utilizing a known solution to determine unknown coefficients, and applying factoring techniques or root relationships to find the remaining solution.

Estimated Time:1m 30s

Question 2468Question →

In rectangle $PQRS$ , the length of side $PQ$ is 8 inches and the length of diagonal $PR$ is 10 inches. What is the length, in inches, of side $QR$ ?

2

6

6.4

12.8

18

Show answer & explanation

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 2469Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 - 6x + 4y - 12 = 0$ . A line $L$ passes through the center of this circle and is perpendicular to the line with equation $3x - 4y = 12$ . What is the $y$ -coordinate of the intersection point of line $L$ and the circle that has a positive $y$ -value?

2

6

-6

1

5

Show answer & explanation

Answer: 2

Answer

The y-coordinate of the intersection point with a positive y-value is 2.

Completing the square of the circle equation

x^2 + y^2 - 6x + 4y - 12 = 0

yields

(x-3)^2 + (y+2)^2 = 25

, giving center

(3, -2)

and radius

r = 5

. The given line

3x - 4y = 12

has a slope of

3/4

, so a perpendicular line

L

has a slope of

-4/3

. Since line

L

passes through the center

(3, -2)

and has a slope of

-4/3

, moving a distance of 5 units along this line (which corresponds to a horizontal change of

\pm 3

and vertical change of

\mp 4

) gives the intersection points

(6, -6)

and

(0, 2)

. The y-coordinate of the point with a positive y-value is 2.

Step-by-Step Solution

1

Complete the square for the circle's equation to find its center and radius.

The equation is rewritten as

(x-3)^2 + (y+2)^2 = 25

, which represents a circle with center

(3, -2)

and radius

r = 5

.

Converting to standard form

(x-h)^2 + (y-k)^2 = r^2

gives the circle's center and radius.

2

Find the slope of the given line

3x - 4y = 12

and determine the slope of the perpendicular line

L

.

The slope of the given line is

3/4

. The slope of line

L

is the negative reciprocal:

-4/3

.

Perpendicular lines have slopes that are negative reciprocals.

3

Find the intersection points of line

L

and the circle.

Since line

L

passes through the center

(3, -2)

and has a slope of

-4/3

, points on the line at a distance of the radius

r=5

are found by moving 3 units horizontally and 4 units vertically. This yields the points

(3 + 3, -2 - 4) = (6, -6)

and

(3 - 3, -2 + 4) = (0, 2)

.

The intersection points lie exactly one radius away from the center along the line.

4

Identify the y-coordinate with a positive value.

Between the points

(6, -6)

and

(0, 2)

, the point with a positive y-value is

(0, 2)

, so the y-coordinate is 2.

The question specifies the intersection point must have a positive y-value.

Key Concept

Circle standard equations and perpendicular lines on the coordinate plane

Question 2470Question →

In $\triangle ABC$ , the measure of exterior angle $\angle ACD$ is $135^\circ$ , where $D$ lies on the extension of side $BC$ past $C$ . If the measure of interior angle $\angle A$ is $25^\circ$ greater than the measure of interior angle $\angle B$ , what is the measure, in degrees, of $\angle B$ ?

Show answer & explanation

Answer: 55

Answer

55

According to the Exterior Angle Theorem, the measure of exterior angle

\angle ACD

is equal to the sum of the measures of its remote interior angles,

\angle A

and

\angle B

. This gives the equation

\text{m}\angle A + \text{m}\angle B = 135^\circ

. Using the information that

\text{m}\angle A = \text{m}\angle B + 25^\circ

, we substitute this expression into the equation to get

(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ

. Simplifying this equation gives

2\text{m}\angle B + 25 = 135

, which simplifies to

2\text{m}\angle B = 110

, and dividing by 2 yields

\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.

\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.

(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ

The problem states that the measure of interior angle

\angle A

is

25^\circ

greater than the measure of interior angle

\angle B

.

3

Solve the algebraic equation for the measure of interior angle

\angle B

.

\text{m}\angle B = 55^\circ

Combining like terms gives

2\text{m}\angle B + 25 = 135

. Subtracting 25 from both sides gives

2\text{m}\angle B = 110

. Dividing by 2 yields

\text{m}\angle B = 55^\circ

.

Key Concept

Exterior Angle Theorem and remote interior angles relation

Question 2471Question →

A convex polygon has $n$ sides. The sum of the measures of its interior angles is $3$ times the sum of the measures of its exterior angles. What is the value of $n$ ?

Show answer & explanation

Answer: 8

Answer

8

The sum of the interior angle measures of an

n

-sided convex polygon is

(n-2) \times 180^\circ

. The sum of the exterior angle measures is always

360^\circ

. According to the problem, the sum of the interior angles is

3

times the sum of the exterior angles, which can be written as the equation

(n-2) \times 180 = 3 \times 360

. Simplifying the right side gives

(n-2) \times 180 = 1080

. Dividing both sides by

180

results in

n - 2 = 6

. Adding

2

to both sides yields

n = 8

.

Step-by-Step Solution

1

Use the formula for the sum of the interior angle measures of a convex polygon with

n

sides.

The sum of the interior angles is

(n-2) \times 180^\circ

.

By the polygon interior angle sum theorem, the sum of the interior angles of any convex

n

-gon is

(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^\circ

.

The sum of the exterior angles of any convex polygon is always constant and equals

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) \times 180^\circ = 3 \times 360^\circ

The problem states that the sum of the interior angles is

3

times the sum of the exterior angles.

4

Solve the equation for

n

.

n = 8

Divide both sides by

180^\circ

to get

n - 2 = 6

, then add

2

to both sides to find

n = 8

.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and the sum of its exterior angles is

360^\circ

.

Question 2472Question →

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

3

5

6

7

8

Show answer & explanation

Answer: 7

Answer

The polygon has 7 sides.

The correct answer is

7

because the sum of the interior angles of a convex polygon with

n

sides is

(n - 2) \times 180^\circ

. Setting this equal to

900^\circ

yields

(n - 2) \times 180 = 900

. Dividing both sides by

180

gives

n - 2 = 5

, and adding

2

to both sides results in

n = 7

.

Step-by-Step Solution

1

Set up the equation for the sum of the interior angles of a polygon.

(n - 2) \times 180^\circ = 900^\circ

The sum of the interior angles of a convex polygon with

n

sides is always

(n - 2) \times 180^\circ

.

2

Divide both sides of the equation by

180^\circ

.

n - 2 = 5

To isolate the term with

n

, we divide

900

by

180

.

3

Solve for

n

by adding

2

to both sides.

n = 7

Adding

2

to both sides isolates the variable

n

.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n - 2) \times 180^\circ

.

Estimated Time:45s

Question 2473Question →

What is the greatest integer value of $k$ that satisfies the inequality $8 - 3(2k - 5) \geq 4(k + 6)$ ?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of

k

that satisfies the inequality is

-1

.

Solving the inequality step-by-step yields

k \leq -0.1

. The greatest integer less than or equal to

-0.1

is

-1

.

Step-by-Step Solution

1

Distribute the coefficients to remove parentheses

8 - 6k + 15 \geq 4k + 24

Expanding the terms makes it possible to combine like terms on each side of the inequality.

2

Combine like terms on the left side

23 - 6k \geq 4k + 24

Simplifying the constant values on the left side (

8 + 15 = 23

) simplifies the expression.

3

Subtract

4k

from both sides

23 - 10k \geq 24

This groups all the variable terms on the left-hand side.

4

Subtract

23

from both sides

-10k \geq 1

This isolates the variable term on the left-hand side.

5

Divide by

-10

and flip the inequality sign

k \leq -0.1

Dividing both sides by a negative number requires reversing the direction of the inequality sign.

6

Identify the greatest integer satisfying the inequality

k = -1

The largest integer that is less than or equal to

-0.1

is

-1

.

Key Concept

Solving linear inequalities and applying the sign-flip rule when dividing by a negative number.

Estimated Time:1m 30s

Question 2474Question →

A triangle has two sides of length $9$ and $14$ . The length of the third side, $s$ , is a multiple of $4$ . What is the sum of all possible integer values for $s$ ?

48

60

56

76

80

Show answer & explanation

Answer: 56

Answer

56

According to the Triangle Inequality Theorem, the length of the third side,

s

, of a triangle with sides of length 9 and 14 must satisfy the inequality

14 - 9 < s < 14 + 9

, which simplifies to

5 < 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,

s

, must satisfy the inequality:

14 - 9 < s < 14 + 9

, which simplifies to

5 < 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)

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 = 56

.

The question asks for the sum of all possible integer values for the third side length

s

.

Key Concept

Triangle Inequality Theorem

Estimated Time:1m 0s

Question 2475Question →

A regular decagon has $10$ sides of equal length. What is the measure, in degrees, of one exterior angle of this decagon?

Show answer & explanation

Answer: 36

Answer

The correct answer is 36.

The sum of the exterior angles of any convex polygon is always

360^\circ

. A regular decagon has

10

congruent sides and therefore

10

congruent exterior angles. Dividing

360^\circ

by

10

yields

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^\circ

. For a regular polygon with

n

sides, the measure of each exterior angle is

\frac{360^\circ}{n}

.

Question 2476Question →

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

40^\circ

60^\circ

80^\circ

108^\circ

150^\circ

Show answer & explanation

Answer: $60^\circ$

Answer

60^\circ

The sum of the interior angles of a convex pentagon is calculated as

(5 - 2) \times 180^\circ = 540^\circ

. Representing the five angle measures in the given ratio as

2x

,

3x

,

4x

,

4x

, and

5x

, their sum is

18x = 540^\circ

, which gives

x = 30^\circ

. The smallest angle corresponds to the smallest term in the ratio, which is

2x = 2 \times 30^\circ = 60^\circ

. Therefore, the correct measure is

60^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles of a convex pentagon.

The sum is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

The sum of the interior angles of any convex

n

-gon is given by the formula

(n - 2) \times 180^\circ

, and a pentagon has

5

sides.

2

Set up an algebraic equation using the given ratio of the angle measures.

Let the angle measures be

2x

,

3x

,

4x

,

4x

, and

5x

. Their sum is

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

x

.

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 \times 30^\circ = 60^\circ

.

The smallest term in the ratio is

2

, so multiplying this term by

x

gives the smallest angle measure.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(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 2477Question →

A geometric sequence consists of positive terms and has a first term of $12$ and a common ratio of $r$ . An arithmetic sequence has a first term of $5$ and a common difference of $d$ . If the $3\text{rd}$ term of the geometric sequence is $3$ and the $4\text{th}$ term of the arithmetic sequence is $7$ , what is the value of $r + d$ ?

\frac{3}{5}

\frac{11}{8}

\frac{7}{6}

-\frac{15}{2}

\frac{35}{48}

Show answer & explanation

Answer: $\frac{7}{6}$

Answer

\frac{7}{6}

The correct value is

\frac{7}{6}

. This is found by first calculating the common ratio of the geometric sequence, where

12r^2 = 3 \Rightarrow r^2 = \frac{1}{4} \Rightarrow r = \frac{1}{2}

, and the common difference of the arithmetic sequence, where

5 + 3d = 7 \Rightarrow 3d = 2 \Rightarrow d = \frac{2}{3}

. Adding these two fractions with a common denominator yields

\frac{3}{6} + \frac{4}{6} = \frac{7}{6}

.

Step-by-Step Solution

1

Find the common ratio

r

of the geometric sequence.

r = \frac{1}{2}

The formula for the

n\text{th}

term of a geometric sequence is

g_n = g_1 \cdot r^{n-1}

. For the

3\text{rd}

term,

g_3 = 12r^2 = 3

, which simplifies to

r^2 = \frac{1}{4}

. Since the sequence has positive terms, we take the positive square root to get

r = \frac{1}{2}

.

2

Find the common difference

d

of the arithmetic sequence.

d = \frac{2}{3}

The formula for the

n\text{th}

term of an arithmetic sequence is

a_n = a_1 + (n-1)d

. For the

4\text{th}

term,

a_4 = 5 + 3d = 7

, which simplifies to

3d = 2

, or

d = \frac{2}{3}

.

3

Calculate the sum of

r

and

d

.

r + d = \frac{7}{6}

Adding the two values with a common denominator of

6

gives

\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}

.

Key Concept

Arithmetic and Geometric Sequences and Series

Estimated Time:1m 30s

Question 2478Question →

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

True

False

Show answer & explanation

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 2479Question →

In the standard $(x, y)$ coordinate plane, a circle is centered at the origin $(0,0)$ and has a radius of $8$ . A horizontal chord $AB$ lies entirely in the first and second quadrants at a distance of $4$ units from the $x$ -axis. A point $P$ is located on the circle such that $\triangle ABP$ is a right triangle. If the hypotenuse of $\triangle ABP$ is a diameter of the circle, what is the area of $\triangle ABP$ ?

16\sqrt{3}

32

32\sqrt{3}

32\sqrt{5}

48\sqrt{3}

Show answer & explanation

Answer: $32\sqrt{3}$

Answer

The area of the right triangle is

32\sqrt{3}

.

The correct answer is

32\sqrt{3}

. The horizontal chord

AB

has

y

-coordinate

4

, and its endpoints lie on the circle

x^2 + y^2 = 64

. Solving for

x

gives

x = \pm 4\sqrt{3}

, so the length of the chord is

8\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

16

). Since

AB < 16

,

AB

is a leg, and the hypotenuse is one of the other sides (e.g.,

AP

). The remaining leg

BP

is found using the Pythagorean theorem:

BP = \sqrt{16^2 - (8\sqrt{3})^2} = \sqrt{256 - 192} = \sqrt{64} = 8

. The area of the right triangle is

\frac{1}{2} \times 8\sqrt{3} \times 8 = 32\sqrt{3}

.

Step-by-Step Solution

1

Determine the length of chord

AB

.

The length of chord

AB

is

8\sqrt{3}

.

The equation of the circle is

x^2 + y^2 = 64

. Since the chord is horizontal and at a distance of

4

units from the

x

-axis, its

y

-coordinate is

4

. Substituting

y = 4

gives

x^2 + 16 = 64 \Rightarrow x^2 = 48 \Rightarrow x = \pm 4\sqrt{3}

. The distance between

A(-4\sqrt{3}, 4)

and

B(4\sqrt{3}, 4)

is

8\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

16

, so the right angle is at

B

(or

A

).

Any right triangle inscribed in a circle must have a diameter as its hypotenuse. The diameter of this circle is

2 \times 8 = 16

. Since the chord

AB = 8\sqrt{3} \approx 13.86

is shorter than the diameter, it cannot be the hypotenuse. Therefore, either

AP

or

BP

is the hypotenuse (a diameter), making the angle opposite to it (either

\angle ABP

or

\angle BAP

) the

90^\circ

angle.

3

Calculate the length of the remaining leg of the right triangle.

The length of leg

BP

is

8

.

Using the Pythagorean theorem for right triangle

ABP

with hypotenuse

AP = 16

and leg

AB = 8\sqrt{3}

:

AB^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

ABP

.

The area is

32\sqrt{3}

.

The area of a right triangle is

\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 2480Question →

A business analyst models a company's weekly net profit, $N(x)$ , in dollars, as the difference between its weekly revenue, $R(x) = (2x + 5)(30 - x)$ , and its weekly production cost, $C(x) = (x - 4)^2 + 150$ , where $x$ represents the number of items sold. When $N(x)$ is simplified and written in standard form as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of the coefficient $b$ ?

Show answer & explanation

Answer: 63

Answer

The coefficient

b

is

63

.

Expanding the revenue polynomial gives

R(x) = -2x^2 + 55x + 150

and the cost polynomial gives

C(x) = x^2 - 8x + 166

. Subtracting them gives

N(x) = (-2x^2 + 55x + 150) - (x^2 - 8x + 166) = -3x^2 + 63x - 16

. The coefficient of the

x

term, which corresponds to

b

, is

63

.

Step-by-Step Solution

1

Expand the revenue expression

R(x) = (2x + 5)(30 - x)

using polynomial multiplication.

R(x) = -2x^2 + 55x + 150

To find the polynomial representing revenue in standard form.

2

Expand the cost expression

C(x) = (x - 4)^2 + 150

using binomial squaring.

C(x) = x^2 - 8x + 166

To find the polynomial representing cost in standard form.

3

Subtract the cost polynomial from the revenue polynomial:

N(x) = R(x) - C(x)

, distributing the negative sign through all terms of the cost polynomial.

N(x) = -3x^2 + 63x - 16

To determine the net profit polynomial

N(x)

in standard form.

4

Identify the coefficient

b

of the

x

term in the standard form

ax^2 + bx + c

.

b = 63

To answer the specific question asking for the coefficient of the middle term.

Key Concept

Polynomial subtraction and expansion of algebraic expressions.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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