All practice questions

Question 2501Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(2, 5)$ and $(6, -3)$ . A second line, $L_2$ , is perpendicular to $L_1$ and intersects $L_1$ at its $y$ -intercept. What is the $x$ -coordinate of the $x$ -intercept of $L_2$ ?

Show answer & explanation

Answer: -18

Answer

The

x

-coordinate of the

x

-intercept of

L_2

is

-18

.

First, the slope of

L_1

is calculated as

-2

using the slope formula. Substituting one of the points into the slope-intercept form gives the

y

-intercept of

L_1

as

(0, 9)

. Since

L_2

is perpendicular to

L_1

, its slope is the negative reciprocal of

-2

, which is

\frac{1}{2}

. Since

L_2

shares the

y

-intercept

(0, 9)

, its equation is

y = \frac{1}{2}x + 9

. Setting

y = 0

to find the

x

-intercept gives

x = -18

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

.

The slope of

L_1

is

-2

.

Using the slope formula

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

with the given points

(2, 5)

and

(6, -3)

yields

m_1 = \frac{-3 - 5}{6 - 2} = -2

.

2

Find the

y

-intercept of

L_1

.

The

y

-intercept is

(0, 9)

.

Substituting

m = -2

and the coordinates of

(2, 5)

into the slope-intercept equation

y = mx + b

gives

5 = -2(2) + b

, which simplifies to

b = 9

.

3

Find the slope of the perpendicular line,

L_2

.

The slope of

L_2

is

\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of

-2

is

\frac{1}{2}

.

4

Determine the equation of

L_2

and calculate its

x

-intercept.

The

x

-coordinate of the

x

-intercept of

L_2

is

-18

.

Since

L_2

passes through the

y

-intercept

(0, 9)

, its equation is

y = \frac{1}{2}x + 9

. Setting

y = 0

to find the

x

-intercept yields

0 = \frac{1}{2}x + 9

, which solves to

x = -18

.

Key Concept

Determining the equation and intercepts of a line perpendicular to a given line that passes through a specific shared point.

Estimated Time:1m 30s

Question 2502Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints $A(-3, k)$ and $B(5, 3)$ . If the midpoint of segment $AB$ lies on the $x$ -axis, what is the length of segment $AB$ ?

Show answer & explanation

Answer: 10

Answer

The length of segment

AB

is 10.

The midpoint of segment

AB

with endpoints

A(-3, k)

and

B(5, 3)

is

\left(1, \frac{k + 3}{2}\right)

. Since the midpoint lies on the

x

-axis, its

y

-coordinate must be

0

. Solving

\frac{k + 3}{2} = 0

gives

k = -3

. This means the endpoints are

A(-3, -3)

and

B(5, 3)

. The distance between these two points is

\sqrt{(5 - (-3))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Set up the equation for the

y

-coordinate of the midpoint.

\frac{k + 3}{2} = 0

The midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

has a

y

-coordinate of

\frac{y_1 + y_2}{2}

. Since the midpoint lies on the

x

-axis, its

y

-coordinate must be

0

.

2

Solve the equation for

k

.

k = -3

Multiply both sides of the equation by

2

to get

k + 3 = 0

, then subtract

3

from both sides.

3

Substitute

k = -3

to find the coordinates of point

A

.

A(-3, -3)

This provides both complete endpoints,

A(-3, -3)

and

B(5, 3)

, which are needed to find the distance.

4

Apply the distance formula to find the length of segment

AB

.

AB = 10

The distance formula is

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Substituting the coordinates gives

AB = \sqrt{(5 - (-3))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

.

Key Concept

Distance and Midpoint Formulas

Question 2503Question →

A geometric sequence of positive terms has a first term of $9$ and a third term of $4$ . What is the sum of the first $4$ terms of this sequence?

\frac{1}{3}

\frac{27}{4}

19

21

\frac{65}{3}

Show answer & explanation

Answer: $\frac{65}{3}$

Answer

The correct answer is

\frac{65}{3}

.

To find the sum of the first 4 terms of a geometric sequence with

a_1 = 9

and

a_3 = 4

, we first determine the common ratio

r

. Since

a_3 = a_1 \cdot r^2

, we have

4 = 9 \cdot r^2

, which gives

r^2 = \frac{4}{9}

. Because the terms are positive,

r = \frac{2}{3}

. The first 4 terms are

9

,

6

,

4

, and

\frac{8}{3}

. Summing these terms gives

9 + 6 + 4 + \frac{8}{3} = 19 + \frac{8}{3} = \frac{65}{3}

.

Step-by-Step Solution

1

Find the common ratio

r

of the geometric sequence.

r = \frac{2}{3}

Since the sequence is geometric, the third term is related to the first term by

a_3 = a_1 \cdot r^2

. Substituting the given values yields

4 = 9 \cdot r^2

, which simplifies to

r^2 = \frac{4}{9}

. Because all terms in the sequence are positive,

r

must be positive, so

r = \sqrt{\frac{4}{9}} = \frac{2}{3}

.

2

Calculate the first 4 terms of the sequence.

a_1 = 9

,

a_2 = 6

,

a_3 = 4

,

a_4 = \frac{8}{3}

Multiply each term by the common ratio

r = \frac{2}{3}

to find the subsequent term:

a_1 = 9

,

a_2 = 9 \cdot \frac{2}{3} = 6

,

a_3 = 6 \cdot \frac{2}{3} = 4

, and

a_4 = 4 \cdot \frac{2}{3} = \frac{8}{3}

.

3

Sum the first 4 terms of the sequence.

Sum =

\frac{65}{3}

Add the four terms:

9 + 6 + 4 + \frac{8}{3} = 19 + \frac{8}{3} = \frac{57}{3} + \frac{8}{3} = \frac{65}{3}

.

Key Concept

Calculating the sum of the first

n

terms of a geometric sequence given its first and third terms.

Question 2504Question →

In isosceles trapezoid $ABCD$ , the parallel bases are $AB$ and $CD$ . If the measure of interior angle $A$ is $70^\circ$ , what is the measure, in degrees, of interior angle $C$ ?

Show answer & explanation

Answer: 110

Answer

The measure of interior angle

C

is

110

degrees.

Since the trapezoid is isosceles with bases

AB

and

CD

, the base angles

\angle A

and

\angle B

are congruent, so

\angle B = 70^\circ

. The consecutive interior angles along the leg

BC

are supplementary because

AB \parallel CD

, which means

\angle B + \angle C = 180^\circ

. Solving for

\angle C

gives

180^\circ - 70^\circ = 110^\circ

.

Step-by-Step Solution

1

Find the measure of angle

B

using the properties of an isosceles trapezoid.

\angle B = 70^\circ

In an isosceles trapezoid, the angles sharing a base are congruent. Since

AB

is a base,

\angle A = \angle B = 70^\circ

.

2

Calculate the measure of angle

C

using the parallel lines property.

\angle C = 110^\circ

Because the bases

AB

and

CD

are parallel, the consecutive interior angles

\angle B

and

\angle C

must sum to

180^\circ

. Therefore,

\angle C = 180^\circ - 70^\circ = 110^\circ

.

Key Concept

Properties of an isosceles trapezoid

Alternative Method

Since the sum of interior angles in any quadrilateral is

360^\circ

, and in an isosceles trapezoid the base angles are equal (

\angle A = \angle B = 70^\circ

and

\angle C = \angle D

), we can write

70^\circ + 70^\circ + \angle C + \angle D = 360^\circ

. Since

\angle C = \angle D

, this simplifies to

140^\circ + 2\angle C = 360^\circ \implies 2\angle C = 220^\circ \implies \angle C = 110^\circ

.

Estimated Time:45s

Question 2505Question →

In the standard $(x, y)$ coordinate plane, quadrilateral $ABCD$ is an isosceles trapezoid where $AB$ is parallel to $CD$ and the length of $AD$ equals the length of $BC$ . The coordinates of three of the vertices are $A(0, 0)$ , $B(16, 12)$ , and $C(9, 13)$ . If $ABCD$ is not a parallelogram, what is the $y$ -coordinate of vertex $D$ ?

-1

1

5

7

13

Show answer & explanation

Answer: $7$

Answer

The correct

y

-coordinate of vertex

D

is

7

.

The correct

y

-coordinate is

7

. The slope of base

AB

is

\frac{12-0}{16-0} = \frac{3}{4}

, meaning the parallel line containing base

CD

is

y = \frac{3}{4}x + \frac{25}{4}

. The length of the leg

BC

is

\sqrt{(9-16)^2 + (13-12)^2} = \sqrt{50}

. Setting the distance of leg

AD

equal to

\sqrt{50}

gives the equation

x^2 + y^2 = 50

. Substituting the line equation yields

x^2 + (\frac{3x+25}{4})^2 = 50

, which simplifies to the quadratic

x^2 + 6x - 7 = 0

. Solving for

x

gives

x = 1

or

x = -7

. If

x = -7

, then

y = 1

, which makes

ABCD

a parallelogram. If

x = 1

, then

y = 7

, which successfully forms a non-parallelogram isosceles trapezoid.

Step-by-Step Solution

1

Calculate the slope of the base

AB

.

Slope

m = \frac{12 - 0}{16 - 0} = \frac{3}{4}

Since the trapezoid has bases

AB \parallel CD

, the line containing segment

CD

must also have a slope of

\frac{3}{4}

.

2

Determine the equation of the line containing segment

CD

passing through

C(9, 13)

.

y - 13 = \frac{3}{4}(x - 9) \implies y = \frac{3}{4}x + \frac{25}{4}

Using the point-slope form with vertex

C

and the parallel slope allows us to express the coordinates of

D

as

(x, \frac{3}{4}x + \frac{25}{4})

.

3

Calculate the length of the leg

BC

using the distance formula.

BC = \sqrt{(9 - 16)^2 + (13 - 12)^2} = \sqrt{(-7)^2 + 1^2} = \sqrt{50}

Since the trapezoid is isosceles with

AD = BC

, the distance from the origin

A(0,0)

to vertex

D(x, y)

must also satisfy

AD = \sqrt{50}

.

4

Set up the distance equation for

AD

and substitute the line equation for

y

.

x^2 + y^2 = 50 \implies x^2 + \left(\frac{3}{4}x + \frac{25}{4}\right)^2 = 50

Substituting the relation for

y

in terms of

x

allows us to solve for the

x

-coordinate of vertex

D

.

5

Solve the quadratic equation for

x

.

16x^2 + (9x^2 + 150x + 625) = 800 \implies 25x^2 + 150x - 175 = 0 \implies x^2 + 6x - 7 = 0 \implies (x + 7)(x - 1) = 0 \implies x = 1 \text{ or } x = -7

The solutions to this quadratic equation yield two potential coordinates for vertex

D

.

6

Find the corresponding

y

-coordinates and verify the non-parallelogram condition.

If

x = -7

, then

y = 1

, which makes

ABCD

a parallelogram. If

x = 1

, then

y = 7

, which makes

ABCD

a non-parallelogram isosceles trapezoid.

The question specifies that

ABCD

is not a parallelogram, so we choose the solution

D(1, 7)

, giving a

y

-coordinate of

7

.

Key Concept

Using coordinate geometry (slopes and distances) to determine the properties and vertices of a quadrilateral.

Alternative Method

In an isosceles trapezoid, the perpendicular bisector of the base is the axis of symmetry. The midpoint of base

AB

is

M(8, 6)

. Since the slope of

AB

is

\frac{3}{4}

, the slope of the perpendicular bisector is the negative reciprocal,

-\frac{4}{3}

. The equation of this perpendicular bisector is

y - 6 = -\frac{4}{3}(x - 8) \implies 4x + 3y - 50 = 0

. Reflecting vertex

C(9, 13)

across this line yields vertex

D

. The projection of

C

onto the bisector is found by intersecting it with the parallel base line

3x - 4y + 25 = 0

, giving the intersection point

P(5, 10)

. Reflecting

C

across

P

gives

D = 2P - C = (2(5) - 9, 2(10) - 13) = (1, 7)

, which confirms the

y

-coordinate is

7

.

Estimated Time:3m 0s

Question 2506Question →

The sum of the measures of all but one of the interior angles of a convex polygon is $2010^\circ$ . What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 150

Answer

The measure of the remaining interior angle is 150 degrees.

The sum of the interior angles of any convex polygon with

n

sides is a multiple of

180^\circ

given by

(n-2) \times 180^\circ

. Because the polygon is convex, the measure of the remaining angle must be strictly less than

180^\circ

. Thus, the total sum of all interior angles must be the smallest multiple of

180^\circ

that is strictly greater than the given sum of

2010^\circ

. Since

11 \times 180^\circ = 1980^\circ

(which is less than

2010^\circ

), the total sum must be at least

12 \times 180^\circ = 2160^\circ

. Subtracting the given sum of the other angles from this total gives

2160^\circ - 2010^\circ = 150^\circ

. Since

150^\circ

is less than

180^\circ

, this is a mathematically valid remaining angle for a convex polygon.

Step-by-Step Solution

1

Set up the inequality for the sum of the interior angles.

The total sum

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

must satisfy

2010^\circ < S < 2010^\circ + 180^\circ

, which simplifies to

2010^\circ < S < 2190^\circ

.

Since the polygon is convex, the remaining interior angle must have a measure strictly between

0^\circ

and

180^\circ

.

2

Determine the value of

n-2

by finding the unique integer multiple.

Dividing the inequality by

180^\circ

gives

11.17 < n-2 < 12.17

. Since

n

must be an integer,

n-2 = 12

, which means the polygon has

n = 14

sides.

The number of sides of a polygon must be a whole number, so

n-2

must be an integer.

3

Calculate the measure of the remaining interior angle.

x = (12 \times 180^\circ) - 2010^\circ = 2160^\circ - 2010^\circ = 150^\circ

.

Subtract the sum of the other interior angles from the total sum of the interior angles of a 14-gon.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and each interior angle of a convex polygon must measure strictly less than

180^\circ

.

Question 2507Question →

A designer is creating a custom floor tile in the shape of an irregular convex pentagon. The tile has one right angle. The remaining four interior angles are in the ratio $2:3:3:4$ . What is the measure, in degrees, of the largest interior angle of this tile?

112.5^\circ

120^\circ

150^\circ

180^\circ

270^\circ

Show answer & explanation

Answer: $150^\circ$

Answer

The largest interior angle of the tile measures

150^\circ

.

The correct answer is

150^\circ

. The sum of the interior angles of a pentagon is

540^\circ

. Subtracting the right angle (

90^\circ

) leaves

450^\circ

for the remaining four angles. Since they are in the ratio

2:3:3:4

, their sum can be represented as

12y = 450^\circ

, which yields

y = 37.5^\circ

. The largest of these angles is

4y = 150^\circ

, which is greater than the other angles (

75^\circ

,

112.5^\circ

, and

90^\circ

).

Step-by-Step Solution

1

Calculate the sum of the interior angles of a pentagon.

The sum of the interior angles is

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

.

A pentagon has 5 sides, and the sum of the interior angles of any convex

n

-gon is given by the formula

(n - 2) \times 180^\circ

.

2

Subtract the right angle to find the sum of the remaining four interior angles.

The sum of the remaining angles is

540^\circ - 90^\circ = 450^\circ

.

One of the angles is a right angle, which measures

90^\circ

.

3

Set up an equation using the given ratio to find the value of one part of the ratio.

Let the four remaining angles be

2y

,

3y

,

3y

, and

4y

. Their sum is

2y + 3y + 3y + 4y = 12y = 450^\circ

, which gives

y = 37.5^\circ

.

The remaining angles are in the ratio

2:3:3:4

, so their measures are proportional to these values.

4

Calculate the measure of the largest interior angle.

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

4y

. Thus, the largest angle is

4 \times 37.5^\circ = 150^\circ

.

The question asks for the measure of the largest interior angle.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratios can be used to partition a total quantity into proportional parts.

Question 2508Question →

A convex polygon has $n$ sides. The sum of the measures of all but one of its interior angles is $2020^\circ$ . What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 140

Answer

The measure of the remaining interior angle is

140^\circ

.

For a convex polygon with

n

sides, the sum of all interior angles is

(n-2) \times 180^\circ

. If we represent the remaining interior angle as

x

, then the sum of all interior angles can be written as

2020^\circ + x

. Because the polygon is convex, the measure of the remaining interior angle must satisfy the inequality

0^\circ < x < 180^\circ

. Substituting this into the sum expression gives the inequality

2020^\circ < (n-2) \times 180^\circ < 2200^\circ

. Dividing by

180^\circ

, we find

11.22 < n-2 < 12.22

. Since

n-2

must be an integer,

n-2

must equal

12

, which means the total sum of the interior angles is

12 \times 180^\circ = 2160^\circ

. The remaining angle is found by subtracting

2020^\circ

from

2160^\circ

, resulting in

140^\circ

.

Step-by-Step Solution

1

Use the polygon interior angle sum formula for an

n

-sided polygon.

The sum of all interior angles is

(n-2) \times 180^\circ

.

This formula connects the number of sides of a convex polygon to the total sum of its interior angles.

2

Set up an equation containing the sum of all but one angle (

2020^\circ

) and the remaining angle (

x

).

(n-2) \times 180^\circ = 2020^\circ + x

The total sum of all interior angles is equal to the sum of the

n-1

known angles plus the remaining angle.

3

Apply the convexity constraint

0^\circ < x < 180^\circ

to construct an inequality for the total sum of the interior angles.

2020^\circ < (n-2) \times 180^\circ < 2200^\circ

Since the remaining angle must be strictly between

0^\circ

and

180^\circ

for a convex polygon, adding

2020^\circ

gives the boundaries for the total sum.

4

Divide the inequality by

180^\circ

to isolate the term

n-2

.

11.22 < n-2 < 12.22

This determines the numerical boundaries for the integer value of

n-2

.

5

Find the unique integer value for

n-2

and calculate the exact total sum of the interior angles.

n-2 = 12

, which gives a total sum of

12 \times 180^\circ = 2160^\circ

.

Because

n

must be an integer,

n-2

must be an integer. The only integer in the interval

(11.22, 12.22)

is

12

.

6

Subtract the sum of the other angles from the total sum of the interior angles to solve for

x

.

x = 2160^\circ - 2020^\circ = 140^\circ

.

This yields the exact value of the remaining interior angle.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

, where each interior angle is strictly between

0^\circ

and

180^\circ

.

Question 2509Question →

A chemical solution in a laboratory has an initial volume of $50$ milliliters and evaporates at a rate of $\frac{7}{2}$ milliliters per hour. A second chemical solution has an initial volume of $20$ milliliters and evaporates at a rate of $\frac{5}{4}$ milliliters per hour. After how many hours, $t$ , will the volume of the first solution be at most the volume of the second solution?

t \ge \frac{40}{3}

t \le \frac{40}{3}

t \ge 90

t \le 90

t \le -\frac{40}{3}

Show answer & explanation

Answer: $t \ge \frac{40}{3}$

Answer

The correct solution is the set of values

t \ge \frac{40}{3}

.

Translating the problem statement gives the inequality

50 - \frac{7}{2}t \le 20 - \frac{5}{4}t

. Subtracting 50 from both sides and adding

\frac{5}{4}t

to both sides results in

-\frac{9}{4}t \le -30

. Multiplying both sides by

-\frac{4}{9}

and reversing the inequality sign because of the negative multiplier yields the correct solution,

t \ge \frac{40}{3}

.

Step-by-Step Solution

1

Write the inequality representing the physical situation.

50 - \frac{7}{2}t \le 20 - \frac{5}{4}t

The volume of the first solution after

t

hours is

50 - \frac{7}{2}t

, and the volume of the second solution is

20 - \frac{5}{4}t

. The phrase 'at most' means the first volume must be less than or equal to the second volume.

2

Isolate the variable terms on the left side and the constant terms on the right side.

-\frac{7}{2}t + \frac{5}{4}t \le 20 - 50

Grouping like terms makes it possible to simplify both sides of the inequality.

3

Find a common denominator of 4 to combine the fractions on the left side, and simplify the constant terms on the right side.

-\frac{9}{4}t \le -30

Converting

-\frac{7}{2}t

to

-\frac{14}{4}t

allows us to add it to

\frac{5}{4}t

, resulting in

-\frac{9}{4}t

.

4

Multiply both sides of the inequality by

-\frac{4}{9}

to solve for

t

, reversing the inequality sign because we are multiplying by a negative number.

t \ge \frac{40}{3}

Multiplying by the reciprocal of the coefficient isolates

t

. The inequality sign must be reversed (

{\le}

to

{\ge}

) because we are multiplying by a negative value.

Key Concept

Solving Linear Inequalities

Estimated Time:1m 30s

Question 2510Question →

In the standard $(x, y)$ coordinate plane, a rhombus $ABCD$ has vertices $A(1, 2)$ and $C(7, 10)$ . The length of diagonal $BD$ is half the length of diagonal $AC$ . If the $x$ -coordinate of vertex $B$ is greater than the $x$ -coordinate of vertex $D$ , what is the $y$ -coordinate of vertex $B$ ?

Show answer & explanation

Answer: 4.5

Answer

The

y

-coordinate of vertex

B

is

4.5

.

By using the geometric properties of a rhombus, we know that its diagonals bisect each other perpendicularly. The midpoint of diagonal

AC

is calculated as

M(4, 6)

and its length is

10

. Consequently, the perpendicular diagonal

BD

must pass through

M(4, 6)

with a slope of

-\frac{3}{4}

(the negative reciprocal of the slope of

AC

, which is

\frac{4}{3}

). Since the length of

BD

is half the length of

AC

, the length of

BD

is

5

, meaning vertices

B

and

D

are each a distance of

2.5

units away from

M(4, 6)

. Solving for points along the line

y - 6 = -0.75(x - 4)

at this distance gives

(6, 4.5)

and

(2, 7.5)

. The condition that the

x

-coordinate of

B

is greater than the

x

-coordinate of

D

uniquely determines

B

to be

(6, 4.5)

, yielding a

y

-coordinate of

4.5

.

Step-by-Step Solution

1

Calculate the midpoint

M

and the length of diagonal

AC

.

M = (4, 6)

and

AC = 10

.

The diagonals of a rhombus bisect each other at their midpoint and their lengths determine the proportions of the shape.

2

Find the slope and length of diagonal

BD

.

Slope of

BD

is

-\frac{3}{4}

, and length is

5

.

Diagonals of a rhombus are perpendicular, meaning their slopes are negative reciprocals (

m_1 \cdot m_2 = -1

). The problem specifies that the length of

BD

is half of

AC

(

10 \div 2 = 5

).

3

Set up equations to find coordinates of

B(x, y)

and

D(x, y)

that are at distance

2.5

from

M(4, 6)

along the line of diagonal

BD

.

(x - 4)^2 + (y - 6)^2 = 6.25

and

y - 6 = -0.75(x - 4)

.

Since the diagonals bisect each other, the distance from the midpoint

M

to each of the remaining vertices

B

and

D

is half the length of diagonal

BD

(

5 \div 2 = 2.5

).

4

Solve the system of equations for the coordinates.

P_1(6, 4.5)

and

P_2(2, 7.5)

.

Substituting

y-6

into the distance equation yields

(x-4)^2 + 0.5625(x-4)^2 = 6.25

, which simplifies to

1.5625(x-4)^2 = 6.25 \implies (x-4)^2 = 4 \implies x - 4 = \pm 2

. Thus,

x_1 = 6

(giving

y_1 = 4.5

) and

x_2 = 2

(giving

y_2 = 7.5

).

5

Identify vertex

B

using the given coordinate condition.

B = (6, 4.5)

, so the

y

-coordinate is

4.5

.

The problem states that the

x

-coordinate of

B

is greater than the

x

-coordinate of

D

. Comparing the two solutions, the one with the larger

x

-value (

6 > 2

) must belong to vertex

B

.

Key Concept

Rhombus Diagonal Properties in the Coordinate Plane

Alternative Method

Alternatively, since the diagonals of a rhombus divide it into four congruent right triangles, we can determine the side length of the rhombus. The legs of these right triangles are half the diagonal lengths:

5

and

2.5

. By the Pythagorean theorem, the square of the side length is

5^2 + 2.5^2 = 31.25

. We can set up distance equations from

B(x, y)

to

A(1, 2)

and

C(7, 10)

:

(x-1)^2 + (y-2)^2 = 31.25

and

(x-7)^2 + (y-10)^2 = 31.25

. Subtracting the second equation from the first simplifies to the linear relation

y = -0.75x + 9

, which can then be substituted back into one of the quadratic equations to find

x = 6

or

x = 2

, yielding

y = 4.5

or

y = 7.5

.

Estimated Time:3m 0s

Question 2511Question →

The endpoint $A$ of segment $AB$ is located at $(-3, 5)$ in the standard $(x, y)$ coordinate plane. Segment $AB$ is translated 4 units right and 2 units down, and then reflected across the line $y = -x$ . What are the coordinates of the image of endpoint $A$ after these transformations?

(-7, 7)

(-7, -1)

(-3, -1)

(3, 1)

(-1, -3)

Show answer & explanation

Answer: $(-3, -1)$

Answer

The coordinates of the image of endpoint

A

are

(-3, -1)

.

Translating the point

A(-3, 5)

4 units right and 2 units down gives

( -3 + 4, 5 - 2 ) = (1, 3)

. Reflecting the point

(1, 3)

across the line

y = -x

swaps and negates the coordinates, giving

(-3, -1)

.

Step-by-Step Solution

1

Apply the translation of 4 units right and 2 units down to the initial coordinates of point

A(-3, 5)

.

The x-coordinate changes by

+4

and the y-coordinate changes by

-2

:

(-3 + 4, 5 - 2) = (1, 3)

.

Translation shifts the coordinates directly by adding the horizontal change to

x

and subtracting the vertical change from

y

.

2

Apply the reflection across the line

y = -x

to the translated point

(1, 3)

.

Swapping and negating both coordinates of

(1, 3)

yields

(-3, -1)

.

The reflection rule across the line

y = -x

maps any point

(x, y)

to

(-y, -x)

.

Key Concept

Composite transformations in the coordinate plane involving translation and reflection across the line

y = -x

.

Question 2512Question →

A geometric sequence has a first term of $\frac{1}{2}$ and a common ratio of $\frac{1}{4}$ . What is the sum of the first 3 terms of this sequence?

\frac{1}{14}

\frac{13}{16}

\frac{21}{32}

\frac{1}{32}

\frac{9}{128}

Show answer & explanation

Answer: $\frac{21}{32}$

Answer

The sum of the first 3 terms of this sequence is

\frac{21}{32}

.

The sum of the first three terms of a geometric sequence is calculated by finding each individual term and then adding them together. The first term is

\frac{1}{2}

. The second term is obtained by multiplying the first term by the common ratio:

\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}

. The third term is obtained by multiplying the second term by the common ratio:

\frac{1}{8} \times \frac{1}{4} = \frac{1}{32}

. To add these terms, we find a common denominator of 32:

\frac{16}{32} + \frac{4}{32} + \frac{1}{32} = \frac{21}{32}

.

Step-by-Step Solution

1

Identify the first three terms of the geometric sequence using the formula

a_n = a_1 \cdot r^{n-1}

.

The first term

a_1

is given as

\frac{1}{2}

. The second term is

a_2 = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}

. The third term is

a_3 = \frac{1}{8} \cdot \frac{1}{4} = \frac{1}{32}

.

Before calculating the sum, each individual term to be summed must be determined.

2

Find a common denominator to add the three fractional terms.

The least common multiple of the denominators 2, 8, and 32 is 32. Express the terms with this common denominator:

a_1 = \frac{16}{32}

,

a_2 = \frac{4}{32}

, and

a_3 = \frac{1}{32}

.

Adding fractions requires a common denominator.

3

Sum the adjusted fractions.

\frac{16}{32} + \frac{4}{32} + \frac{1}{32} = \frac{21}{32}

.

This yields the total sum of the first three terms.

Key Concept

Calculating the sum of a finite geometric series by finding and summing individual terms.

Question 2513Question →

A landscape architect designs a square courtyard with a side length of $4x - 3$ meters. A square garden bed with a side length of $2x - 1$ meters is placed in one of the corners of the courtyard. The remaining area of the courtyard is paved. If the paved area, in square meters, is represented by the polynomial $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $b$ ?

Show answer & explanation

Answer: -20

Answer

The value of

b

is

-20

.

The remaining paved area is the difference between the area of the courtyard and the garden bed:

(16x^2 - 24x + 9) - (4x^2 - 4x + 1) = 12x^2 - 20x + 8

. The coefficient of the linear

x

term is

-20

.

Step-by-Step Solution

1

Find the polynomial representing the total area of the courtyard.

A_{\text{total}} = (4x - 3)^2 = 16x^2 - 24x + 9

square meters

The area of a square is the square of its side length. We expand

(4x - 3)^2

using the identity

(p - q)^2 = p^2 - 2pq + q^2

.

2

Find the polynomial representing the area of the garden bed.

A_{\text{garden}} = (2x - 1)^2 = 4x^2 - 4x + 1

square meters

The garden bed is also a square, so we expand

(2x - 1)^2

using the binomial squaring identity.

3

Subtract the garden bed area from the total area to find the paved area.

A_{\text{paved}} = 12x^2 - 20x + 8

square meters

The paved area is the difference between the two areas. We distribute the negative sign to all terms of the subtracted polynomial:

-(4x^2 - 4x + 1) = -4x^2 + 4x - 1

, and then combine like terms.

4

Identify the coefficient of the

x

term,

b

.

b = -20

Comparing

12x^2 - 20x + 8

to

ax^2 + bx + c

, the coefficient of the linear

x

term is

-20

.

Key Concept

Operations on Polynomials

Alternative Method

To find only the coefficient of the linear term, expand and subtract only the linear terms from both binomial expansions:

2(4x)(-3) - 2(2x)(-1) = -24x - (-4x) = -20x

. The coefficient

b

is therefore

-20

.

Estimated Time:1m 30s

Question 2514Question →

In the figure, line $L_1$ is parallel to line $L_2$ . Vertex $A$ of $\triangle ABC$ lies on $L_1$ , and vertices $B$ and $C$ lie on $L_2$ . Side $AB$ is perpendicular to $L_2$ . Point $D$ lies on $L_2$ such that $C$ is between $B$ and $D$ . If the measure of the exterior angle $\angle ACD$ is $132^\circ$ , what is the measure, in degrees, of $\angle BAC$ ?

Show answer & explanation

Answer: 42

Answer

42

The correct answer is

42

. Because

AB

is perpendicular to

L_2

,

\angle ABC

is

90^\circ

. The exterior angle

\angle ACD

is given as

132^\circ

, which means the adjacent interior angle

\angle ACB

must be supplementary to it:

180^\circ - 132^\circ = 48^\circ

. Since the interior angles of a triangle always sum to

180^\circ

, the remaining angle

\angle BAC

is

180^\circ - (90^\circ + 48^\circ) = 42^\circ

. Alternatively, applying the Exterior Angle Theorem, the exterior angle is equal to the sum of the two remote interior angles:

\angle ACD = \angle ABC + \angle BAC

, so

132^\circ = 90^\circ + \angle BAC

, which simplifies to

\angle BAC = 42^\circ

.

Step-by-Step Solution

1

Determine the measure of interior angle

\angle ABC

.

\angle ABC = 90^\circ

Since side

AB

is perpendicular to

L_2

, the angle it makes with

L_2

at vertex

B

is

90^\circ

.

2

Find the measure of interior angle

\angle ACB

.

\angle ACB = 48^\circ

The interior angle

\angle ACB

and the exterior angle

\angle ACD

form a linear pair along line

L_2

, making them supplementary:

\angle ACB = 180^\circ - 132^\circ = 48^\circ

.

3

Calculate the measure of

\angle BAC

using the angle sum of a triangle.

42^\circ

The interior angles of

\triangle ABC

sum to

180^\circ

. Subtracting the known angles gives

\angle BAC = 180^\circ - (90^\circ + 48^\circ) = 42^\circ

.

Key Concept

Triangle Angle Sum Theorem and Supplementary Angle Relationships

Question 2515Question →

In the standard $(x, y)$ coordinate plane, a triangle has a vertex at $T(-1, 4)$ . The triangle is reflected across the $y$ -axis, and then dilated by a scale factor of $2$ with the center of dilation at the point $(3, 1)$ . What are the coordinates of the image of vertex $T$ after this sequence of transformations?

(5, 7)

(2, 8)

(-1, 7)

(-5, -9)

(7, -5)

Show answer & explanation

Answer: $(-1, 7)$

Answer

The correct coordinates of the image of vertex

T

are

(-1, 7)

.

The correct coordinates are found by first reflecting the vertex across the

y

-axis to get

(1, 4)

, and then applying the dilation formula relative to the center

(3, 1)

with a scale factor of

2

, which yields

(-1, 7)

.

Step-by-Step Solution

1

Reflect the vertex

T(-1, 4)

across the

y

-axis.

T'(1, 4)

Reflecting a point

(x, y)

across the

y

-axis maps it to

(-x, y)

, which changes the sign of the x-coordinate of

T(-1, 4)

to get

(1, 4)

.

2

Apply the dilation with a scale factor of

2

centered at

(3, 1)

to the reflected point

T'(1, 4)

.

T''(-1, 7)

The dilation formula for a point

(x, y)

with scale factor

k

and center

(h, k_c)

is

(h + k(x - h), k_c + k(y - k_c))

. Substituting

x = 1

,

y = 4

,

k = 2

,

h = 3

, and

k_c = 1

yields the x-coordinate

3 + 2(1 - 3) = -1

and the y-coordinate

1 + 2(4 - 1) = 7

.

Key Concept

Composite transformations in the coordinate plane involving reflections and non-origin-centered dilations.

Alternative Method

Instead of calculating algebraically, you can sketch the transformations on a grid. Plot

T(-1, 4)

, reflect it across the y-axis to find

T'(1, 4)

. Mark the center of dilation at

P(3, 1)

. Find the vector from

P

to

T'

, which is

2

units left and

3

units up. Double this vector to

4

units left and

6

units up from

P(3, 1)

, which lands at

(-1, 7)

.

Estimated Time:1m 30s

Question 2516Question →

In rhombus $ABCD$ , the diagonals $AC$ and $BD$ intersect at point $E$ . If the length of segment $AE$ is $3$ inches and the length of segment $BE$ is $4$ inches, what is the perimeter, in inches, of the rhombus?

5

14

20

28

100

Show answer & explanation

Answer: 20

Answer

The perimeter of the rhombus is 20 inches.

The diagonals of a rhombus are perpendicular and bisect each other, forming four right triangles at their intersection. In right triangle

AEB

, the legs are

AE = 3

inches and

BE = 4

inches. Using the Pythagorean theorem, the hypotenuse (which is the side

AB

of the rhombus) is

\sqrt{3^2 + 4^2} = 5

inches. Because a rhombus has four sides of equal length, the perimeter is

4 \times 5 = 20

inches.

Step-by-Step Solution

1

Identify the properties of the diagonals of a rhombus.

The diagonals of a rhombus are perpendicular bisectors of each other. This means they intersect at a

90^\circ

angle and divide each other into equal halves.

This allows us to model the relationship between the diagonals and the sides using right triangles.

2

Calculate the side length of the rhombus using the Pythagorean theorem.

In the right triangle

AEB

formed by the intersection of the diagonals, the legs are

AE = 3

inches and

BE = 4

inches. The side

AB

is the hypotenuse:

AB = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

inches.

Knowing the side length is necessary to find the perimeter of the rhombus.

3

Calculate the perimeter of the rhombus.

Since all four sides of a rhombus are equal in length, the perimeter is

4 \times 5 = 20

inches.

The perimeter of any polygon is the sum of its outer boundary lengths.

Key Concept

Properties of Rhombus Diagonals and Perimeter

Question 2517Question →

Consider two lines in a coordinate plane: Line $A$ , represented by the equation $3x + ky = 8$ for some constant $k$ , and Line $B$ , which contains the points $(2, -1)$ and $(5, 8)$ . If Line $A$ is perpendicular to Line $B$ , what is the value of $k$ ?

9

-9

1

-1

3

Show answer & explanation

Answer: 9

Answer

The value of

k

is

9

.

The correct answer is the value

9

. First, we find the slope of Line

B

using the coordinate points

(2, -1)

and

(5, 8)

, which gives a slope of

3

. Since Line

A

is perpendicular to Line

B

, its slope must be the negative reciprocal of

3

, which is

-\frac{1}{3}

. By rewriting the equation of Line

A

,

3x + ky = 8

, in slope-intercept form, we identify its slope as

-\frac{3}{k}

. Setting these two slope values equal gives

-\frac{3}{k} = -\frac{1}{3}

, which solves to

k = 9

.

Step-by-Step Solution

1

Calculate the slope of Line

B

using the slope formula

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

with the points

(2, -1)

and

(5, 8)

.

The slope of Line

B

is

m_B = \frac{8 - (-1)}{5 - 2} = \frac{9}{3} = 3

.

Finding the slope of the given line is necessary to determine the perpendicular slope.

2

Determine the perpendicular slope of Line

A

using the relationship

m_A = -\frac{1}{m_B}

.

The slope of Line

A

must be

m_A = -\frac{1}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Express the slope of Line

A

in terms of

k

by rewriting

3x + ky = 8

in slope-intercept form (

y = mx + b

).

Subtracting

3x

from both sides gives

ky = -3x + 8

, and dividing by

k

yields

y = -\frac{3}{k}x + \frac{8}{k}

. The slope is

-\frac{3}{k}

.

This allows us to set up an equation to solve for

k

.

4

Equate the slope of Line

A

to the perpendicular slope found in Step 2 and solve for

k

.

-\frac{3}{k} = -\frac{1}{3} \Rightarrow \frac{3}{k} = \frac{1}{3} \Rightarrow k = 9

.

Solving this equation gives the value of

k

that makes the lines perpendicular.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

Estimated Time:1m 30s

Question 2518Question →

A right triangle has a hypotenuse of length $x + 4$ inches. The lengths of the two legs of the triangle are $x$ inches and $x + 2$ inches. What is the value of $x$ ?

2

6

8

10

12

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6 because applying the Pythagorean theorem yields the relation

x^2 + (x + 2)^2 = (x + 4)^2

. Expanding the binomials gives

x^2 + x^2 + 4x + 4 = x^2 + 8x + 16

, which simplifies to the quadratic equation

x^2 - 4x - 12 = 0

. Factoring this equation yields

(x - 6)(x + 2) = 0

. Discarding the negative solution

x = -2

because length must be positive leaves the correct solution of 6.

Step-by-Step Solution

1

Set up the equation using the Pythagorean theorem, where the sum of the squares of the legs equals the square of the hypotenuse.

x^2 + (x + 2)^2 = (x + 4)^2

The sides of a right triangle must satisfy the Pythagorean relation

a^2 + b^2 = c^2

.

2

Expand the squared binomial terms on both sides of the equation.

x^2 + (x^2 + 4x + 4) = x^2 + 8x + 16

Applying the algebraic identity

(a+b)^2 = a^2 + 2ab + b^2

is necessary to simplify the terms.

3

Simplify the equation and move all terms to one side to set the quadratic expression to zero.

x^2 - 4x - 12 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve quadratic equations.

4

Factor the quadratic equation.

(x - 6)(x + 2) = 0

Factoring allows finding the roots by setting each linear binomial factor to zero.

5

Solve for

x

and discard any physically impossible negative values.

x = 6

(since

x = -2

is discarded)

A physical measurement of side length must be strictly positive.

Key Concept

Formulating and solving quadratic equations derived from the Pythagorean theorem by expanding binomials and factoring.

Question 2519Question →

A designer is creating a custom tiled floor using irregular convex pentagonal tiles. In each pentagon, the measures of three of the interior angles are in the ratio $2:3:4$ . The other two interior angles are congruent to each other, and each is $15^\circ$ less than the sum of the two smallest angles in the ratio. What is the measure, in degrees, of the largest interior angle of one of these pentagonal tiles?

120^\circ

135^\circ

150^\circ

165^\circ

105^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

The correct answer is 135 degrees. The largest interior angle of the pentagon is one of the two congruent angles.

The correct answer is 135 degrees. The sum of the interior angles of a pentagon is 540 degrees. Representing the three angles in the ratio as 2x, 3x, and 4x gives a sum of 9x. The remaining two angles are each equal to the sum of the two smallest ratio terms minus 15, which is 5x - 15. The sum of all five angles is 19x - 30 = 540, which yields x = 30. Evaluating the angles gives 60, 90, 120, 135, and 135 degrees. The largest of these is 135 degrees.

Step-by-Step Solution

1

Determine the sum of the interior angles of a pentagon.

The sum is

540^\circ

.

The formula for the sum of the interior angles of an

n

-sided polygon is

(n-2) \times 180^\circ

. For a pentagon (

n = 5

), the sum is

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

.

2

Set up an algebraic equation representing the sum of all five interior angles.

The equation is

2x + 3x + 4x + 2(5x - 15^\circ) = 540^\circ

.

Let the three angles in the ratio

2:3:4

be represented as

2x

,

3x

, and

4x

. The sum of the two smallest is

2x + 3x = 5x

. Each of the other two congruent angles is

15^\circ

less than this sum, which is

5x - 15^\circ

.

3

Solve the algebraic equation for

x

.

x = 30^\circ

.

Simplify the equation:

9x + 10x - 30^\circ = 540^\circ \implies 19x = 570^\circ \implies x = 30^\circ

.

4

Calculate the measures of all five interior angles and identify the largest.

The angles are

60^\circ

,

90^\circ

,

120^\circ

,

135^\circ

, and

135^\circ

. The largest angle is

135^\circ

.

Substitute

x = 30^\circ

into each expression:

2(30) = 60^\circ

,

3(30) = 90^\circ

,

4(30) = 120^\circ

, and

5(30) - 15 = 135^\circ

for the other two. The largest value among these is

135^\circ

.

Key Concept

Polygon interior angle sum and algebraic representation of ratios

Question 2520Question →

For a certain convex polygon with $n$ sides, the sum of the measures of its interior angles is exactly $24$ times the measure of one exterior angle of a regular polygon with $n$ sides. If this polygon is regular, what is the measure, in degrees, of each of its interior angles?

45

120

135

165

1080

Show answer & explanation

Answer: 135

Answer

135

By translating the problem description into an algebraic relationship, we establish that the sum of the interior angles,

(n-2) \times 180^\circ

, equals

24

times the measure of a single exterior angle,

\frac{360^\circ}{n}

. Simplifying this equation by dividing both sides by

180

yields

n - 2 = \frac{48}{n}

. Multiplying by

n

creates the quadratic equation

n^2 - 2n - 48 = 0

, which factors as

(n-8)(n+6) = 0

. Since the number of sides of a polygon must be positive,

n = 8

. For a regular octagon (

n=8

), the measure of each interior angle is

\frac{(8-2) \times 180^\circ}{8} = 135^\circ

.

Step-by-Step Solution

1

Set up the equation based on the geometric properties of polygons.

(n-2) \times 180 = 24 \times \frac{360}{n}

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. The measure of one exterior angle of a regular polygon with

n

sides is

\frac{360^\circ}{n}

.

2

Simplify the equation and solve for the number of sides

n

.

n^2 - 2n - 48 = 0

, which gives

n = 8

.

Dividing both sides by

180

simplifies the equation to

n - 2 = \frac{48}{n}

. Multiplying by

n

yields the quadratic equation

n(n - 2) = 48

, or

n^2 - 2n - 48 = 0

. Factoring gives

(n - 8)(n + 6) = 0

. Since the number of sides must be a positive integer,

n = 8

.

3

Calculate the measure of each interior angle of a regular octagon (

n = 8

).

135^\circ

The measure of each interior angle of a regular polygon is given by

\frac{(n-2) \times 180^\circ}{n}

. Substituting

n = 8

yields

\frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ

.

Key Concept

The relationship between the sum of interior angles, the number of sides, and the exterior angles of regular polygons.

Estimated Time:2m 30s

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