Plane Geometry Practice Questions — ACT — Page 5

Question 81Question →

The ratio of the measure of an interior angle of a regular polygon to the measure of its exterior angle is $3:1$ . How many sides does this polygon have?

4

6

8

10

12

Show answer & explanation

Answer: 8

Answer

8

The interior angle and exterior angle of a polygon at any vertex are supplementary, meaning they add up to

180^\circ

. Given the ratio of the interior angle to the exterior angle is

3:1

, we can express their measures as

3x

and

x

. Solving the equation

3x + x = 180^\circ

gives

4x = 180^\circ

, which means

x = 45^\circ

. Therefore, the measure of each exterior angle of the regular polygon is

45^\circ

. Since the sum of the exterior angles of any convex polygon is always

360^\circ

, the number of sides

n

is calculated by dividing

360^\circ

by the measure of one exterior angle:

n = \frac{360^\circ}{45^\circ} = 8

.

Step-by-Step Solution

1

Set up an equation for the interior and exterior angles using the given ratio.

Let the measure of the exterior angle be

x

and the measure of the interior angle be

3x

.

The ratio of the interior angle to the exterior angle is

3:1

, so their measures can be represented as

3x

and

x

respectively.

2

Use the fact that the interior angle and exterior angle at any vertex of a polygon are supplementary (form a linear pair).

3x + x = 180^\circ \implies 4x = 180^\circ \implies x = 45^\circ

.

An interior angle and its adjacent exterior angle always lie on a straight line and sum to

180^\circ

.

3

Calculate the number of sides of the regular polygon using the measure of one exterior angle.

n = \frac{360^\circ}{45^\circ} = 8

.

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

360^\circ

. For a regular polygon with

n

sides, each exterior angle measures

\frac{360^\circ}{n}

.

Key Concept

The relationship between the interior and exterior angles of a regular polygon, and the formula relating the number of sides to the sum of the exterior angles.

Question 82Question →

A regular hexagon $ABCDEF$ has a side length of $8$ inches. Point $M$ lies on side $CD$ such that the length of segment $CM$ is $2$ inches. What is the length, in inches, of segment $AM$ ?

Show answer & explanation

Answer: 14

Answer

14

The correct answer is 14 because

\triangle ACM

is a right triangle with legs

AC = 8\sqrt{3}

and

CM = 2

. Applying the Pythagorean Theorem yields

AM^2 = (8\sqrt{3})^2 + 2^2 = 192 + 4 = 196

, so

AM = \sqrt{196} = 14

.

Step-by-Step Solution

1

Find the properties of the regular hexagon and the diagonal

AC

.

The interior angle at vertex

B

is

120^\circ

. Since

AB = BC = 8

, the triangle

\triangle ABC

is an isosceles triangle with angles

\angle BAC = \angle BCA = 30^\circ

. Using the properties of

30^\circ

-

60^\circ

-

90^\circ

triangles, the diagonal length is

AC = 8\sqrt{3}

.

To find the length of the leg

AC

for the right triangle

\triangle ACM

.

2

Determine the angle

\angle ACD

to show

\triangle ACM

is a right triangle.

Since the interior angle

\angle BCD = 120^\circ

and

\angle BCA = 30^\circ

, the remaining angle is

\angle ACD = 120^\circ - 30^\circ = 90^\circ

. Thus,

\triangle ACM

is a right triangle with the right angle at vertex

C

.

To establish the right-angle relationship between the legs

AC

and

CM

.

3

Apply the Pythagorean Theorem to calculate the hypotenuse

AM

.

AM^2 = AC^2 + CM^2 = (8\sqrt{3})^2 + 2^2 = 192 + 4 = 196

. Taking the square root gives

AM = 14

.

To find the final length of segment

AM

.

Key Concept

Using properties of regular hexagons, special right triangles, and the Pythagorean Theorem to find lengths in multi-step plane geometry configurations.

Question 83Question →

A triangle has side lengths of $x$ , $2x$ , and $15$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

7

8

9

10

11

Show answer & explanation

Answer: 9

Answer

9

Applying the Triangle Inequality Theorem yields three inequalities:

x + 2x > 15

(which simplifies to

x > 5

),

x + 15 > 2x

(which simplifies to

x < 15

), and

2x + 15 > x

(which is always true since

x

is positive). Combining these results in

5 < x < 15

. The integers in this range are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

. Counting them gives 9 possible values.

Step-by-Step Solution

1

Set up the three inequalities using the Triangle Inequality Theorem for the sides

x

,

2x

, and

15

.

The inequalities are

x + 2x > 15

,

x + 15 > 2x

, and

2x + 15 > x

.

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

2

Solve each inequality for

x

.

3x > 15 \implies x > 5

, and

15 > x \implies x < 15

. The third inequality

x > -15

is always true for positive side lengths.

To find the range of valid values for

x

.

3

Combine the inequalities and count the possible integer values for

x

.

The combined range is

5 < x < 15

. The integer values are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

, which gives

14 - 6 + 1 = 9

values.

Since

x

is specified as an integer, we must identify and count all integers strictly between 5 and 15.

Key Concept

Triangle Inequality Theorem

Alternative Method

Instead of algebraic manipulation, we can test values of

x

directly. For

x=5

, the sides are

5, 10, 15

, but

5+10=15

, which does not form a triangle. For

x=15

, the sides are

15, 30, 15

, but

15+15=30

, which also does not form a triangle. Testing integer values between

5

and

15

confirms they all satisfy the triangle inequality, resulting in 9 valid integers.

Estimated Time:1m 0s

Question 84Question →

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 85Question →

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 86Question →

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 87Question →

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 88Question →

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 89Question →

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 90Question →

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 91Question →

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 92Question →

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 93Question →

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

Question 94Question →

A security camera is mounted on a vertical wall at point $C$ , exactly $15$ feet above the flat ground. The camera is programmed to monitor two objects, $A$ and $B$ , on the ground. The line of sight from the camera to object $A$ makes a $30^\circ$ angle with the wall, and the line of sight from the camera to object $B$ makes a $45^\circ$ angle with the wall. On the ground, the path from the base of the wall directly below the camera to object $A$ is perpendicular to the path from the base of the wall to object $B$ . What is the straight-line distance, in feet, between object $A$ and object $B$ ?

5\sqrt{6}

4\sqrt{15}

10\sqrt{3}

15 + 5\sqrt{3}

30

Show answer & explanation

Answer: $10\sqrt{3}$

Answer

The distance between object

A

and object

B

is

10\sqrt{3}

feet.

The horizontal distances from the base of the wall to objects

A

and

B

are calculated using the trigonometric ratios of the

30^\circ

-

60^\circ

-

90^\circ

and

45^\circ

-

45^\circ

-

90^\circ

triangles formed by the vertical wall. This gives legs of

5\sqrt{3}

feet and

15

feet. Applying the Pythagorean theorem to the right triangle on the ground yields a hypotenuse of

10\sqrt{3}

feet.

Step-by-Step Solution

1

Find the horizontal distance from the base of the wall to object

A

.

Let

O

be the base of the wall directly below the camera

C

, so

OC = 15

feet. The line of sight

CA

makes a

30^\circ

angle with the wall, so

\triangle OCA

is a

30^\circ

-

60^\circ

-

90^\circ

right triangle with leg

OA = 15 \tan(30^\circ) = \frac{15}{\sqrt{3}} = 5\sqrt{3}

feet.

We need to determine the length of one of the perpendicular legs on the ground.

2

Find the horizontal distance from the base of the wall to object

B

.

The line of sight

CB

makes a

45^\circ

angle with the wall, so

\triangle OCB

is a

45^\circ

-

45^\circ

-

90^\circ

right triangle with leg

OB = 15 \tan(45^\circ) = 15

feet.

We need to determine the length of the other perpendicular leg on the ground.

3

Use the Pythagorean theorem to calculate the straight-line distance

AB

on the ground.

Since the paths

OA

and

OB

are perpendicular,

\triangle AOB

is a right triangle with legs

OA = 5\sqrt{3}

and

OB = 15

. The hypotenuse

AB

is

\sqrt{(5\sqrt{3})^2 + 15^2} = \sqrt{75 + 225} = \sqrt{300} = 10\sqrt{3}

feet.

The straight-line distance between the two objects corresponds to the hypotenuse of the right triangle formed by their ground distances.

Key Concept

Applying special right triangle ratios and the Pythagorean theorem to solve multi-step problems in three-dimensional contexts.

Question 95Question →

A geometry student claims that every rectangle is also a parallelogram. Is this claim true or false?

True

False

Show answer & explanation

Answer: True

Answer

True

The claim is true because a rectangle must have four right angles, which mathematically forces opposite sides to be parallel, thereby meeting the definition of a parallelogram.

Step-by-Step Solution

1

Define a rectangle based on its angle properties.

A rectangle is a quadrilateral with four right angles.

This is the primary defining characteristic of a rectangle.

2

Define a parallelogram based on its side properties.

A parallelogram is a quadrilateral with two pairs of parallel opposite sides.

This is the primary defining characteristic of a parallelogram.

3

Analyze the relationship between the two definitions.

In a quadrilateral with four

90^\circ

angles, consecutive angles sum to

180^\circ

. By the consecutive interior angles converse, the opposite sides are parallel. Since both pairs of opposite sides are parallel, the quadrilateral must be a parallelogram. Thus, every rectangle is a parallelogram.

To determine whether the student's statement is correct.

Key Concept

Hierarchical classification of quadrilaterals

Estimated Time:30s

Question 96Question →

In $\triangle ABC$ , the lengths of the sides are $AB = 8$ , $BC = 11$ , and $AC = 14$ . Which of the following inequalities correctly compares the measures of the interior angles of $\triangle ABC$ ?

m\angle A < m\angle B < m\angle C

m\angle B < m\angle A < m\angle C

m\angle A < m\angle C < m\angle B

m\angle C < m\angle A < m\angle B

m\angle B < m\angle C < m\angle A

Show answer & explanation

Answer: $m\angle C < m\angle A < m\angle B$

Answer

The correct inequality is

m\angle C < m\angle A < m\angle B

.

In any triangle, the order of the measures of the interior angles matches the order of the lengths of their opposite sides. Since the side lengths are ordered

AB < BC < AC

(

8 < 11 < 14

), their opposite angles must be ordered in the same way. The angle opposite side

AB

is

\angle C

, the angle opposite side

BC

is

\angle A

, and the angle opposite side

AC

is

\angle B

. Therefore, the correct relationship is

m\angle C < m\angle A < m\angle B

.

Step-by-Step Solution

1

Identify the theorem relating triangle side lengths to their opposite angles.

The Side-Angle Relationship Theorem states that in any triangle, the order of the measures of the angles is the same as the order of the lengths of the sides opposite to those angles.

This establishes the rule needed to compare the angle measures based on the given side lengths.

2

Determine the opposite angle for each side length in

\triangle ABC

.

The angle opposite side

AB

is

\angle C

. The angle opposite side

BC

is

\angle A

. The angle opposite side

AC

is

\angle B

.

This maps each side length to the correct angle it controls.

3

Order the side lengths and apply the mapping to order the angle measures.

Since

8 < 11 < 14

, we write the side inequality as

AB < BC < AC

. Substituting the corresponding opposite angles gives

m\angle C < m\angle A < m\angle B

.

This yields the final correct inequality comparing the angle measures.

Key Concept

Triangle Side-Angle Relationship Theorem

Alternative Method

Another way to solve this is to sketch the triangle to scale. By drawing the longest side

AC = 14

as the horizontal base, the shortest side

AB = 8

on the left, and the side

BC = 11

on the right, it becomes visually apparent that the angle opposite the longest side (

\angle B

at the top vertex) is the largest angle, and the angle opposite the shortest side (

\angle C

at the bottom-right vertex) is the smallest angle.

Estimated Time:1m 0s

Question 97Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the length of $AC$ is $16$ units. Point $D$ lies on side $BC$ such that the measure of $\angle ADB$ is $45^\circ$ . What is the length of segment $CD$ , rounded to the nearest tenth?

Show answer & explanation

Answer: 5.9

Answer

The length of segment CD is approximately 5.9 units.

By recognizing that triangle ABC is a 30-60-90 right triangle, the shorter leg AB is found to be 8 (half of the hypotenuse 16), and the longer leg BC is 8*sqrt(3). Because triangle ABD is a 45-45-90 right triangle, the leg BD is equal to the leg AB, which is 8. Subtracting BD from BC yields CD = 8*sqrt(3) - 8, which is approximately 5.9.

Step-by-Step Solution

1

Determine the type of triangle ABC

Triangle ABC is a 30-60-90 special right triangle.

The triangle has a right angle (90 degrees) at B and an angle of 60 degrees at A, which leaves 30 degrees for angle C.

2

Calculate the lengths of sides AB and BC

AB = 8 units and BC = 8*sqrt(3) units.

In a 30-60-90 triangle with hypotenuse AC = 16, the side opposite 30 degrees (AB) is half the hypotenuse, and the side opposite 60 degrees (BC) is the shorter leg multiplied by sqrt(3).

3

Determine the type of triangle ABD

Triangle ABD is a 45-45-90 special right triangle.

Since D lies on BC, angle ABD is a right angle (90 degrees). Given that angle ADB is 45 degrees, the remaining angle BAD must also be 45 degrees.

4

Calculate the length of side BD

BD = 8 units.

In a 45-45-90 right triangle, the two legs opposite the 45-degree angles are equal in length, so BD = AB.

5

Calculate the length of segment CD and round to the nearest tenth

CD ≈ 5.9 units.

Since D lies on side BC, CD = BC - BD = 8*sqrt(3) - 8 ≈ 8(1.732) - 8 = 13.856 - 8 = 5.856, which rounds to 5.9.

Key Concept

Applying properties of 30-60-90 and 45-45-90 special right triangles to find segment lengths within nested figures.

Question 98Question →

A regular polygon has an interior angle that is $140^\circ$ greater than its exterior angle. How many sides does this polygon have?

8

9

12

18

36

Show answer & explanation

Answer: 18

Answer

The regular polygon has 18 sides.

The correct answer is 18. By setting up the equations for the interior angle

I

and exterior angle

E

, we have

I = E + 140^\circ

and

I + E = 180^\circ

. Substituting the first equation into the second gives

(E + 140^\circ) + E = 180^\circ

, which simplifies to

2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

. The number of sides

n

of a regular polygon is given by

n = 360^\circ / E

. Thus,

n = 360^\circ / 20^\circ = 18

.

Step-by-Step Solution

1

Set up the algebraic relationship between the interior angle (

I

) and the exterior angle (

E

).

I = E + 140^\circ

The problem states that the interior angle is

140^\circ

greater than the exterior angle.

2

Use the supplementary relationship between any interior angle and its corresponding exterior angle.

I + E = 180^\circ

An interior angle and its adjacent exterior angle form a linear pair, which sums to

180^\circ

.

3

Substitute the expression for

I

from Step 1 into the equation from Step 2 and solve for

E

.

(E + 140^\circ) + E = 180^\circ \implies 2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

Substituting allows us to solve for a single variable representing the exterior angle.

4

Calculate the number of sides (

n

) using the sum of the exterior angles of a convex polygon.

n = \frac{360^\circ}{E} = \frac{360^\circ}{20^\circ} = 18

The sum of the exterior angles of any convex polygon is

360^\circ

, and in a regular polygon, all

n

exterior angles are equal.

Key Concept

Relationship between interior and exterior angles of a regular polygon

Estimated Time:1m 30s

Question 99Question →

The interior angles of a quadrilateral are in the ratio $2:3:3:4$ . What is the measure, in degrees, of the largest interior angle of this quadrilateral?

60

90

120

180

240

Show answer & explanation

Answer: 120

Answer

120

The correct answer is 120. The sum of the interior angles of any convex quadrilateral is 360 degrees. Given the ratio of the interior angles is

2:3:3:4

, the total number of parts is

2 + 3 + 3 + 4 = 12

. Dividing the total angle sum by the total parts gives the value of one part:

360^\circ / 12 = 30^\circ

. The largest angle has 4 parts, so its measure is

4 \times 30^\circ = 120^\circ

.

Step-by-Step Solution

1

Calculate the sum of the parts in the given ratio of the angles.

2 + 3 + 3 + 4 = 12

parts

To find the fractional share of each angle, we must first determine the total number of equal parts in the ratio.

2

Determine the value in degrees of a single part of the ratio using the sum of interior angles of a quadrilateral.

360^\circ / 12 = 30^\circ

per part

The sum of the interior angles of any convex quadrilateral is 360 degrees. Dividing this sum by the total number of parts gives the angle measure of one part.

3

Multiply the value of one part by the number of parts of the largest angle.

4 \times 30^\circ = 120^\circ

The largest angle corresponds to the largest number in the ratio, which is 4.

Key Concept

Sum of interior angles of a quadrilateral and ratio distribution

Question 100Question →

In the standard $(x, y)$ coordinate plane, quadrilateral $ABCD$ is a trapezoid with $AB$ parallel to $CD$ . The vertices are $A(2, 9)$ , $B(5, 3)$ , and $C(2, 1)$ . The diagonals $AC$ and $BD$ intersect at point $E$ , and the line segment $BD$ is perpendicular to $AC$ . If the ratio of the area of $\triangle ABE$ to the area of $\triangle CDE$ is $9:1$ , what is the $x$ -coordinate of vertex $D$ ?

-7

-1

1

\frac{5}{3}

3

Show answer & explanation

Answer: $1$

Answer

The correct answer is

1

.

The vertical diagonal

AC

lies on

x = 2

, and the perpendicular diagonal

BD

lies on

y = 3

. Their intersection point is

E(2, 3)

. Since the bases

AB

and

CD

of the trapezoid are parallel, the triangles

\triangle ABE

and

\triangle CDE

formed by the diagonals are similar. The ratio of their areas is

9:1

, which means the ratio of their corresponding sides is the square root of the area ratio, which is

3:1

. The horizontal segment

BE

has a length of

5 - 2 = 3

, meaning the segment

DE

must have a length of

3 \div 3 = 1

. Since

D

must lie to the left of the intersection to keep

CD

parallel to

AB

, its

x

-coordinate is

2 - 1 = 1

.

Step-by-Step Solution

1

Find the equation of the line containing diagonal

AC

.

The line is

x = 2

.

Since vertices

A(2, 9)

and

C(2, 1)

share the same

x

-coordinate of

2

, the diagonal

AC

is a vertical line segment.

2

Find the equation of the line containing diagonal

BD

and locate the intersection

E

.

The line is

y = 3

, and the intersection point is

E(2, 3)

.

Since

BD

is perpendicular to the vertical diagonal

AC

, it must be horizontal. Thus, all points on

BD

share the same

y

-coordinate as

B(5, 3)

. The intersection of

x = 2

and

y = 3

is

E(2, 3)

.

3

Determine the similarity ratio of

\triangle ABE

and

\triangle CDE

.

The ratio of the corresponding sides is

3:1

.

Since

AB \parallel CD

, the alternate interior angles

\angle EAB = \angle ECD

and

\angle EBA = \angle EDC

make

\triangle ABE

similar to

\triangle CDE

. The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths, so

\frac{BE}{DE} = \sqrt{9} = 3

.

4

Calculate the length of

BE

and find the length of

DE

.

The length of

BE

is

3

, and the length of

DE

is

1

.

Using the coordinates of

B(5, 3)

and

E(2, 3)

, the horizontal distance is

BE = 5 - 2 = 3

. Since

\frac{BE}{DE} = 3

, we have

DE = 1

.

5

Determine the coordinates of vertex

D

and verify that

AB \parallel CD

.

The

x

-coordinate of

D

is

1

.

Since

DE = 1

and

D

lies on the line

y = 3

,

D

can be at

(3, 3)

or

(1, 3)

. The slope of

AB

is

\frac{3 - 9}{5 - 2} = -2

. If

D

is

(1, 3)

, the slope of

CD

is

\frac{3 - 1}{1 - 2} = -2

, which is parallel. If

D

is

(3, 3)

, the slope is

2

, which is not parallel. Thus, the

x

-coordinate of

D

must be

1

.

Key Concept

Properties of Quadrilaterals

Estimated Time:3m 0s

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