Plane Geometry Practice Questions — ACT — Page 3

Question 41Question →

In acute triangle $PQR$ , an altitude $PS$ is drawn from vertex $P$ perpendicular to side $QR$ at point $S$ . The measure of $\angle PQS$ is $60^\circ$ , the length of segment $PQ$ is $24$ , and the length of segment $PR$ is $39$ . What is the length of side $QR$ ?

Show answer & explanation

Answer: 45

Answer

The length of side

QR

is

45

.

The altitude divides the acute triangle into two right triangles. In the first right triangle,

\triangle PQS

, the angles are

30^\circ

,

60^\circ

, and

90^\circ

, with a hypotenuse of

24

. This makes the adjacent leg

QS = 12

and the shared altitude

PS = 12\sqrt{3}

. In the second right triangle,

\triangle PRS

, the hypotenuse is

39

and one leg is

12\sqrt{3}

. Using the Pythagorean Theorem, we find the other leg

SR = \sqrt{39^2 - (12\sqrt{3})^2} = \sqrt{1521 - 432} = \sqrt{1089} = 33

. Summing the two segments gives the total length of side

QR = 12 + 33 = 45

.

Step-by-Step Solution

1

Identify the two right triangles formed by the altitude.

The altitude

PS

divides

\triangle PQR

into two adjacent right triangles:

\triangle PQS

and

\triangle PRS

, which share the side

PS

.

Establishing these right triangles allows us to apply right-triangle trigonometric ratios and the Pythagorean Theorem.

2

Use the properties of the

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle

\triangle PQS

to find

QS

and

PS

.

QS = 12

and

PS = 12\sqrt{3}

.

In a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle, the leg opposite the

30^\circ

angle is half the hypotenuse, and the leg opposite the

60^\circ

angle is

\sqrt{3}

times the shorter leg. Here, hypotenuse

PQ = 24

, so

QS = 12

and

PS = 12\sqrt{3}

.

3

Apply the Pythagorean Theorem to

\triangle PRS

to find

SR

.

SR = 33

.

In right triangle

\triangle PRS

, the hypotenuse is

PR = 39

. By the Pythagorean Theorem,

PS^2 + SR^2 = PR^2

. Squaring the sides gives

(12\sqrt{3})^2 + SR^2 = 39^2 \implies 432 + SR^2 = 1521

. Solving for

SR

gives

SR^2 = 1089 \implies SR = 33

.

4

Sum the segments

QS

and

SR

to find the total length of

QR

.

QR = 45

.

Because

\triangle PQR

is an acute triangle, the altitude

PS

lands at a point

S

on the segment

QR

, meaning

QR = QS + SR

. Adding the lengths gives

12 + 33 = 45

.

Key Concept

Applying properties of

30^\circ\text{-}60^\circ\text{-}90^\circ

special right triangles and the Pythagorean Theorem in multi-step geometric figures.

Estimated Time:2m 30s

Question 42Question →

The interior angles of a quadrilateral are in the ratio $2:3:4:6$ . What is the degree measure of the largest interior angle of the quadrilateral?

4

48

72

144

240

Show answer & explanation

Answer: 144

Answer

144

The sum of the interior angles of a quadrilateral is

360^\circ

. Given the ratio

2:3:4:6

, the sum of the parts is

2 + 3 + 4 + 6 = 15

. The value of one part is

360^\circ \div 15 = 24^\circ

. The largest angle corresponds to the largest part of the ratio, which is

6

. Therefore, the largest angle measure is

6 \times 24^\circ = 144^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles of a quadrilateral.

The sum of the interior angles of any quadrilateral is

360^\circ

.

This is a fundamental property of quadrilaterals, which can also be derived using the formula

(n - 2) \times 180^\circ

with

n = 4

.

2

Calculate the total number of parts in the given ratio.

The sum of the ratio parts is

2 + 3 + 4 + 6 = 15

.

Adding the parts of the ratio allows us to find the size of a single share of the total angle sum.

3

Find the measure of one part of the ratio.

One part is equal to

360^\circ \div 15 = 24^\circ

.

Dividing the total sum of the angles by the sum of the ratio parts determines the angle measure per ratio unit.

4

Multiply the largest ratio part by the value of one part to find the largest angle.

The largest angle is

6 \times 24^\circ = 144^\circ

.

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

6

.

Key Concept

Using ratios to find angle measures in a polygon.

Estimated Time:45s

Question 43Question →

In $\triangle ABC$ , the measures of the interior angles $\angle A$ , $\angle B$ , and $\angle C$ are in the ratio $3:4:5$ , respectively. What is the measure of the largest exterior angle of $\triangle ABC$ ?

45^\circ

75^\circ

105^\circ

108^\circ

135^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

135 degrees

The interior angles of a triangle sum to

180^\circ

. Since the angles are in the ratio

3:4:5

, they can be represented as

3x

,

4x

, and

5x

. Adding these gives

12x = 180^\circ

, which simplifies to

x = 15^\circ

. The interior angles are therefore

45^\circ

,

60^\circ

, and

75^\circ

. Because an exterior angle is supplementary to its adjacent interior angle, the largest exterior angle is paired with the smallest interior angle:

180^\circ - 45^\circ = 135^\circ

. Alternatively, the exterior angle is equal to the sum of the two non-adjacent interior angles:

60^\circ + 75^\circ = 135^\circ

.

Step-by-Step Solution

1

Represent the interior angles algebraically using the given ratio.

Let the measures of the interior angles be

3x

,

4x

, and

5x

.

The ratio of the angles is

3:4:5

, so their measures must be multiples of these ratio numbers by the same factor

x

.

2

Set up and solve an equation for

x

using the triangle angle sum theorem.

3x + 4x + 5x = 180^\circ \implies 12x = 180^\circ \implies x = 15^\circ

.

The sum of the interior angles of any triangle is always

180^\circ

.

3

Determine the measures of the three interior angles.

The angles measure

3(15^\circ) = 45^\circ

,

4(15^\circ) = 60^\circ

, and

5(15^\circ) = 75^\circ

.

Multiplying the value of

x

by each term of the ratio gives the individual interior angle measures.

4

Find the largest exterior angle of the triangle.

The largest exterior angle is supplementary to the smallest interior angle:

180^\circ - 45^\circ = 135^\circ

.

An exterior angle and its adjacent interior angle form a linear pair and sum to

180^\circ

. The smallest interior angle will yield the largest exterior angle.

Key Concept

Triangle Angle Sum Theorem and Exterior Angle Relationships

Alternative Method

The exterior angle at any vertex of a triangle is equal to the sum of the measures of the two opposite interior angles. The two largest interior angles are

60^\circ

and

75^\circ

. Therefore, the largest exterior angle is the sum of these two angles:

60^\circ + 75^\circ = 135^\circ

.

Estimated Time:1m 0s

Question 44Question →

In right triangle $ABC$ with $\angle B = 90^\circ$ and $\angle A = 30^\circ$ , the hypotenuse $AC$ has a length of $12$ centimeters. An altitude $BD$ is drawn from vertex $B$ to hypotenuse $AC$ . From point $D$ , a perpendicular segment $DE$ is drawn to side $AB$ , with point $E$ lying on $AB$ . What is the length, in centimeters, of segment $EC$ ?

3\sqrt{7}

\frac{3\sqrt{22}}{2}

\frac{3\sqrt{19}}{2}

\frac{3\sqrt{13}}{2}

\sqrt{39}

Show answer & explanation

Answer: $\frac{3\sqrt{19}}{2}$

Answer

The length of segment EC is \frac{3\sqrt{19}}{2} centimeters.

To find the length of segment EC, we can construct the right triangle EBC with a right angle at B. By using the properties of 30-60-90 right triangles, we determine the side lengths of the triangles in the figure: first finding BC = 6 and AB = 6\sqrt{3} in triangle ABC; then finding AD = 9 in triangle ABD; then finding AE = \frac{9\sqrt{3}}{2} in triangle ADE; and finally finding EB = AB - AE = \frac{3\sqrt{3}}{2}. Applying the Pythagorean theorem to right triangle EBC yields EC = \sqrt{(\frac{3\sqrt{3}}{2})^2 + 6^2} = \frac{3\sqrt{19}}{2}.

Step-by-Step Solution

1

Determine the side lengths of the main right triangle ABC. Since angle A = 30 degrees and angle B = 90 degrees, triangle ABC is a 30-60-90 right triangle. With hypotenuse AC = 12, the leg opposite the 30-degree angle is BC = \frac{12}{2} = 6, and the leg opposite the 60-degree angle is AB = 6\sqrt{3}.

BC = 6 and AB = 6\sqrt{3}

Knowing the side lengths of triangle ABC is necessary to find the dimensions of the smaller inscribed triangles.

2

Find the length of segment AD in right triangle ABD. Altitude BD is perpendicular to AC, making triangle ABD a right triangle with right angle ADB. Since angle A = 30 degrees, triangle ABD is also a 30-60-90 right triangle with hypotenuse AB = 6\sqrt{3}. The side adjacent to the 30-degree angle, AD, is given by AB \times \cos(30^\circ) = 6\sqrt{3} \times \frac{\sqrt{3}}{2} = 9.

AD = 9

Determining AD allows us to analyze the smaller right triangle ADE built on it.

3

Find the lengths of segments AE and EB. In right triangle ADE (where DE is perpendicular to AB), the hypotenuse is AD = 9 and angle A = 30 degrees. The side adjacent to the 30-degree angle is AE = AD \times \cos(30^\circ) = 9 \times \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{2}. Segment EB is then found by subtracting AE from AB: EB = AB - AE = 6\sqrt{3} - \frac{9\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}.

EB = \frac{3\sqrt{3}}{2}

We need the length of segment EB to apply the Pythagorean theorem in the final right triangle EBC.

4

Apply the Pythagorean theorem to right triangle EBC. Since line segment AB is perpendicular to BC, angle EBC is a right angle. In right triangle EBC, the legs are EB = \frac{3\sqrt{3}}{2} and BC = 6. The hypotenuse EC is calculated as EC = \sqrt{EB^2 + BC^2} = \sqrt{(\frac{3\sqrt{3}}{2})^2 + 6^2} = \sqrt{\frac{27}{4} + 36} = \sqrt{\frac{171}{4}} = \frac{3\sqrt{19}}{2}.

EC = \frac{3\sqrt{19}}{2}

Applying the Pythagorean theorem to the legs EB and BC gives the length of the hypotenuse EC.

Key Concept

Applying 30-60-90 right triangle properties and the Pythagorean theorem across multiple connected geometric figures.

Estimated Time:3m 0s

Question 45Question →

A vertical flagpole casts a horizontal shadow on the ground. The distance from the top of the flagpole to the tip of the shadow is $20$ feet. If the length of the shadow is $16$ feet, what is the height, in feet, of the flagpole?

12

16

25.6

36

4

Show answer & explanation

Answer: 12

Answer

12

The flagpole, ground, and line from the top of the pole to the tip of the shadow form a right triangle. The diagonal distance of

20

feet represents the hypotenuse, and the shadow length of

16

feet represents one of the legs. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we set up the equation

a^2 + 16^2 = 20^2

, which simplifies to

a^2 + 256 = 400

. Subtracting

256

from both sides gives

a^2 = 144

. Taking the square root of both sides gives the height of the flagpole as

12

feet.

Step-by-Step Solution

1

Identify the parts of the right triangle formed by the flagpole, ground, and the line from the top of the flagpole to the shadow's tip.

The hypotenuse (

c

) is

20

feet, and one leg (

b

) is

16

feet.

The flagpole is vertical and the ground is horizontal, forming a right angle. The distance from the top of the pole to the tip of the shadow is the diagonal (hypotenuse).

2

Apply the Pythagorean theorem

a^2 + b^2 = c^2

to find the missing leg (

a

).

a^2 + 16^2 = 20^2

, which simplifies to

a^2 + 256 = 400

.

The Pythagorean theorem relates the lengths of the sides of a right triangle.

3

Solve for the unknown height

a

by subtracting and taking the square root.

a^2 = 144 \implies a = 12

feet.

Isolating

a^2

gives

144

, and taking the square root of

144

gives the height of the flagpole.

Key Concept

Pythagorean Theorem

Question 46Question →

In a right triangle, the measure of one of the acute angles is $30^\circ$ . If the side opposite this $30^\circ$ angle has a length of $6.5$ inches, what is the length, in inches, of the hypotenuse?

Show answer & explanation

Answer: 13

Answer

The length of the hypotenuse is

13

inches.

In a

30^\circ

-

60^\circ

-

90^\circ

special right triangle, the length of the hypotenuse is exactly twice the length of the shorter leg (the side opposite the

30^\circ

angle). Given that the shorter leg has a length of

6.5

inches, the hypotenuse has a length of

2 \times 6.5 = 13

inches.

Step-by-Step Solution

1

Determine the relationship between the given side and the hypotenuse using special right triangle properties.

The triangle is a

30^\circ

-

60^\circ

-

90^\circ

right triangle, meaning the hypotenuse is twice the length of the shorter leg.

By geometric theorem, the sides of a

30^\circ

-

60^\circ

-

90^\circ

triangle are in the ratio

1 : \sqrt{3} : 2

, with the shortest side opposite the

30^\circ

angle and the longest side being the hypotenuse.

2

Multiply the length of the side opposite the

30^\circ

angle by 2.

6.5 \text{ inches} \times 2 = 13 \text{ inches}

Doubling the shorter leg length of

6.5

inches gives the length of the hypotenuse.

Key Concept

In a

30^\circ

-

60^\circ

-

90^\circ

special right triangle, the length of the hypotenuse is always twice the length of the shorter leg (the side opposite the

30^\circ

angle).

Question 47Question →

In $\triangle ABC$ , point $D$ lies on side $BC$ such that $AD = BD$ . If the measure of $\angle ADC$ is $112^\circ$ and the measure of $\angle BAC$ is $85^\circ$ , what is the measure of $\angle C$ , in degrees?

Show answer & explanation

Answer: 39

Answer

The measure of

\angle C

is

39^\circ

.

The measure of

\angle C

is found by first calculating the interior angle

\angle ADB = 180^\circ - 112^\circ = 68^\circ

since

BDC

forms a straight line. Because

AD = BD

,

\triangle ABD

is isosceles with

\angle B = \angle BAD

. Using the angle sum of

\triangle ABD

, we have

2(\angle B) + 68^\circ = 180^\circ

, which yields

\angle B = 56^\circ

. Finally, using the angle sum of

\triangle ABC

, we calculate

\angle C = 180^\circ - (85^\circ + 56^\circ) = 39^\circ

.

Step-by-Step Solution

1

Find the measure of

\angle ADB

using the linear pair relationship with

\angle ADC

.

\angle ADB = 68^\circ

Angles on a straight line add up to

180^\circ

. Since

D

lies on

BC

,

\angle ADB + \angle ADC = 180^\circ

.

2

Calculate the measure of

\angle B

using the properties of the isosceles triangle

ABD

.

\angle B = 56^\circ

Since

AD = BD

, the base angles opposite to these sides are equal:

\angle BAD = \angle B

. The sum of angles in

\triangle ABD

is

180^\circ

, so

2(\angle B) + 68^\circ = 180^\circ

.

3

Find the measure of

\angle C

using the triangle angle sum theorem on the large triangle

ABC

.

\angle C = 39^\circ

The sum of the angles in

\triangle ABC

is

180^\circ

, meaning

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

. Substituting the known values gives

85^\circ + 56^\circ + \angle C = 180^\circ

.

Key Concept

Using the Isosceles Triangle Theorem, the Triangle Angle Sum Theorem, and linear pairs to trace unknown angles in a geometric figure.

Practice More

Try finding the missing angles when a transversal cuts two parallel lines that form a triangle with a third intersecting line.

Alternative Method

Instead of finding

\angle B

first and then solving for

\angle C

in

\triangle ABC

, we can find the angle

\angle DAC

first. Since

\angle ADC = 112^\circ

is an exterior angle to

\triangle ABD

, we have

\angle ADC = \angle B + \angle BAD

. Since

\angle B = \angle BAD

, we get

2(\angle BAD) = 112^\circ \implies \angle BAD = 56^\circ

. Because

\angle BAC = 85^\circ

, we have

\angle DAC = 85^\circ - 56^\circ = 29^\circ

. Now looking at

\triangle ADC

, we can solve for

\angle C

directly:

\angle C = 180^\circ - (112^\circ + 29^\circ) = 39^\circ

.

Estimated Time:1m 30s

Question 48Question →

An equilateral triangle $ABC$ has a side length of $12$ inches. An altitude $AD$ is drawn from vertex $A$ to the side $BC$ . A point $P$ lies on the segment $AD$ such that $\triangle BPC$ is a right triangle with a right angle at $P$ . What is the length, in inches, of the segment $AP$ ?

6\sqrt{3} - 3\sqrt{2}

6\sqrt{3} - 6\sqrt{2}

6\sqrt{3} - 6

6\sqrt{3} + 6

12\sqrt{3} - 6

Show answer & explanation

Answer: $6\sqrt{3} - 6$

Answer

The correct answer is

6\sqrt{3} - 6

inches.

The correct answer is

6\sqrt{3} - 6

. Since the side length of the equilateral triangle is

12

, the altitude

AD

splits it into two

30^\circ

-

60^\circ

-

90^\circ

right triangles with base

BD = 6

and altitude

AD = 6\sqrt{3}

. The right triangle

\triangle BPC

has

\angle BPC = 90^\circ

and

PB = PC

, making it an isosceles right triangle. The altitude

PD

splits

\triangle BPC

into two

45^\circ

-

45^\circ

-

90^\circ

right triangles, so

PD = BD = 6

. The length of

AP

is found by subtracting

PD

from

AD

, yielding

6\sqrt{3} - 6

.

Step-by-Step Solution

1

Find the length of the altitude

AD

using the properties of the

30^\circ

-

60^\circ

-

90^\circ

triangle

\triangle ABD

.

The length of

AD

is

6\sqrt{3}

inches.

Since

\triangle ABC

is equilateral with side length

12

inches, the altitude

AD

bisects the base

BC

, making

BD = 6

inches. The altitude splits the equilateral triangle into two

30^\circ

-

60^\circ

-

90^\circ

right triangles. The length of the longer leg is the shorter leg multiplied by

\sqrt{3}

, which gives

AD = 6\sqrt{3}

.

2

Find the length of the segment

PD

using the properties of the

45^\circ

-

45^\circ

-

90^\circ

triangle

\triangle PDB

.

The length of

PD

is

6

inches.

Since

P

lies on the altitude

AD

, which is the axis of symmetry,

\triangle BPC

is an isosceles right triangle with

\angle BPC = 90^\circ

. The altitude

PD

is perpendicular to

BC

and bisects

\angle BPC

, forming two

45^\circ

-

45^\circ

-

90^\circ

right triangles:

\triangle PDB

and

\triangle PDC

. In a

45^\circ

-

45^\circ

-

90^\circ

triangle, the two legs are congruent, so

PD = BD = 6

.

3

Subtract the length of

PD

from the length of

AD

to find the length of segment

AP

.

AP = 6\sqrt{3} - 6

inches.

Since point

P

lies on segment

AD

, the length of

AP

is the difference between the total altitude

AD

and the segment

PD

.

Key Concept

Properties of special right triangles (30-60-90 and 45-45-90) and their multi-step application in geometry.

Estimated Time:2m 0s

Question 49Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $30^\circ$ , and the length of leg $BC$ is $12$ . An altitude $BD$ is drawn perpendicular to the hypotenuse $AC$ . Let $E$ be the midpoint of the altitude $BD$ . A line passing through $E$ is perpendicular to $BD$ and intersects the leg $AB$ at $G$ and the leg $BC$ at $F$ . What is the length of segment $GF$ ?

Show answer & explanation

Answer: 12

Answer

The length of segment

GF

is

12

.

The correct answer is

12

. By analyzing the geometric properties of the 30-60-90 right triangle

ABC

, the altitude

BD

is found to be

6\sqrt{3}

, making the half-segment

BE = 3\sqrt{3}

. The perpendicular line at

E

creates two smaller 30-60-90 right triangles,

\triangle BEF

and

\triangle BEG

. Solving for the legs along the line gives

EF = 3

and

EG = 9

, which sum to

12

.

Step-by-Step Solution

1

Find the length of altitude

BD

in right triangle

ABC

.

BD = 6\sqrt{3}

In right triangle

ABC

, we have

\angle B = 90^\circ

,

\angle A = 30^\circ

, and

\angle C = 60^\circ

. The altitude

BD

forms a smaller 30-60-90 right triangle

BCD

with hypotenuse

BC = 12

. Since

BD

is opposite the

60^\circ

angle

\angle C

, we have

BD = BC \sin(60^\circ) = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}

.

2

Calculate the length of segment

BE

.

BE = 3\sqrt{3}

Since

E

is the midpoint of the altitude

BD

, we divide the length of

BD

by 2:

BE = \frac{6\sqrt{3}}{2} = 3\sqrt{3}

.

3

Determine the length of segment

EF

in right triangle

BEF

.

EF = 3

Since the line

GF

is perpendicular to

BD

,

\angle BEF = 90^\circ

. In right triangle

BCD

, we have

\angle DBC = 30^\circ

, which means

\angle EBF = 30^\circ

. This makes

\triangle BEF

a 30-60-90 right triangle where

BE

is adjacent to the

30^\circ

angle and

EF

is opposite to it. Thus,

EF = \frac{BE}{\sqrt{3}} = \frac{3\sqrt{3}}{\sqrt{3}} = 3

.

4

Determine the length of segment

EG

in right triangle

BEG

.

EG = 9

Since

\angle BEG = 90^\circ

and

\angle ABD = 90^\circ - \angle DBC = 60^\circ

, the angle

\angle EBG = 60^\circ

. This makes

\triangle BEG

a 30-60-90 right triangle where

BE

is adjacent to the

60^\circ

angle and

EG

is opposite to it. Thus,

EG = BE \sqrt{3} = 3\sqrt{3} \times \sqrt{3} = 9

.

5

Calculate the total length of segment

GF

.

GF = 12

Since

G

,

E

, and

F

are collinear and

E

lies between

G

and

F

, the length of segment

GF

is the sum of

EG

and

EF

:

GF = 9 + 3 = 12

.

Key Concept

Using properties of 30-60-90 special right triangles to find segment lengths in complex geometric configurations.

Question 50Question →

In right triangle $DEF$ , the measure of $\angle E$ is $90^\circ$ and the measure of $\angle D$ is $60^\circ$ . If the hypotenuse $DF$ has a length of $14$ centimeters, what is the length, in centimeters, of the segment $DE$ ?

7\sqrt{2}

7\sqrt{3}

7

14\sqrt{3}

28

Show answer & explanation

Answer: $7$

Answer

The length of the segment

DE

is

7

centimeters.

The correct option is the one with the value

7

. In right triangle

DEF

, the angles are

90^\circ

,

60^\circ

, and

30^\circ

, making it a special

30^\circ

-

60^\circ

-

90^\circ

right triangle. The side

DE

is opposite the

30^\circ

angle (the shorter leg). By the properties of a

30^\circ

-

60^\circ

-

90^\circ

triangle, the shorter leg is half the length of the hypotenuse. Thus,

DE = 14 / 2 = 7

centimeters.

Step-by-Step Solution

1

Determine the measure of the third angle,

\angle F

.

\angle F = 30^\circ

The sum of angles in a triangle is

180^\circ

. Since

\angle E = 90^\circ

and

\angle D = 60^\circ

, we calculate

\angle F = 180^\circ - 90^\circ - 60^\circ = 30^\circ

.

2

Identify the relationship between the sides of the

30^\circ

-

60^\circ

-

90^\circ

triangle.

DE

is the shorter leg, opposite

\angle F

(

30^\circ

).

The side opposite the

30^\circ

angle is the shorter leg, which is half the length of the hypotenuse.

3

Calculate the length of

DE

.

DE = 7

centimeters

Since the hypotenuse

DF = 14

centimeters, the shorter leg

DE

is

14 / 2 = 7

centimeters.

Key Concept

In a

30^\circ

-

60^\circ

-

90^\circ

right triangle, the lengths of the sides are in the ratio

1 : \sqrt{3} : 2

. The shorter leg (opposite the

30^\circ

angle) is half the length of the hypotenuse.

Estimated Time:1m 0s

Question 51Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the hypotenuse $AC$ has a length of $20$ centimeters. Point $D$ lies on leg $BC$ such that the length of segment $BD$ is $2\sqrt{3}$ centimeters. A line segment $DE$ is drawn perpendicular to $AC$ such that $E$ lies on $AC$ . What is the length, in centimeters, of segment $AE$ ?

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By solving for the angles and side lengths of the two nested

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangles, we find that the segment

BC

is

10\sqrt{3}

cm, making

DC = 8\sqrt{3}

cm. Using the ratio of sides for the smaller right triangle

DEC

, we find

EC = 12

cm, which leaves

AE = 20 - 12 = 8

cm.

Step-by-Step Solution

1

Determine the third angle of right triangle

ABC

.

\angle C = 30^\circ

The sum of angles in a triangle is

180^\circ

. Since

\angle B = 90^\circ

and

\angle A = 60^\circ

, we have

\angle C = 180^\circ - 90^\circ - 60^\circ = 30^\circ

.

2

Calculate the length of the side

BC

.

BC = 10\sqrt{3}

cm

In the

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle

ABC

, the side

BC

is opposite the

60^\circ

angle, so its length is the hypotenuse

AC

multiplied by

\sin(60^\circ)

or

\frac{\sqrt{3}}{2}

. Thus,

BC = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3}

.

3

Find the length of segment

DC

.

DC = 8\sqrt{3}

cm

Since point

D

lies on segment

BC

, the length of

DC

is the total length of

BC

minus the length of

BD

. Since

BD = 2\sqrt{3}

, we have

DC = 10\sqrt{3} - 2\sqrt{3} = 8\sqrt{3}

.

4

Determine the properties of the right triangle

DEC

.

\triangle DEC

is a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle with hypotenuse

DC = 8\sqrt{3}

cm.

Since segment

DE

is perpendicular to

AC

,

\angle DEC = 90^\circ

. Triangle

DEC

shares the angle

\angle C = 30^\circ

with triangle

ABC

, which makes it a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle where

DC

is the hypotenuse.

5

Calculate the length of segment

EC

.

EC = 12

cm

In the

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle

DEC

, the leg

EC

is adjacent to the

30^\circ

angle, so its length is the hypotenuse

DC

multiplied by

\cos(30^\circ)

or

\frac{\sqrt{3}}{2}

. Thus,

EC = 8\sqrt{3} \times \frac{\sqrt{3}}{2} = 12

.

6

Calculate the length of segment

AE

.

AE = 8

cm

Since point

E

lies on segment

AC

, we can find

AE

by subtracting

EC

from

AC

. Thus,

AE = AC - EC = 20 - 12 = 8

.

Key Concept

Using the properties of

30^\circ\text{-}60^\circ\text{-}90^\circ

special right triangles to find missing lengths in composite geometric configurations.

Question 52Question →

The lengths of the three sides of a triangle are in the ratio $3:4:x$ , where $x$ is an integer. If the perimeter of the triangle is $36$ centimeters, how many different possible values can $x$ have?

1

3

2

4

5

Show answer & explanation

Answer: 2

Answer

There are 2 possible integer values for

x

.

The correct answer is the value of 2. By expressing the side lengths as

3k

,

4k

, and

xk

where

k

must be a positive integer, the perimeter equation becomes

k(7 + x) = 36

. Solving for

7 + x

yields factors of

36

greater than

7

, which are

9, 12, 18,

and

36

. These correspond to

x

values of

2, 5, 11,

and

29

. Checking each set of side lengths against the Triangle Inequality Theorem shows that only the sets corresponding to

x = 2

(sides

12, 16, 8

) and

x = 5

(sides

9, 12, 15

) form valid triangles.

Step-by-Step Solution

1

Define the side lengths using a multiplier

k

.

Let the side lengths of the triangle be

3k

,

4k

, and

xk

for some positive multiplier

k

. Since all three side lengths must be integers, the difference between the first two sides,

4k - 3k = k

, must also be an integer. Thus,

k

must be a positive integer.

This establishes that the scaling factor

k

is a positive integer, allowing us to find discrete solutions.

2

Set up the perimeter equation and express

x

in terms of

k

.

The perimeter is the sum of the side lengths:

3k + 4k + xk = 36 \implies k(7 + x) = 36

. Since

k

and

x

are positive integers,

7 + x

must be a factor of

36

that is greater than

7

.

This constrains the possible values of

x

to the factors of

36

that are larger than

7

.

3

Find the potential values of

x

and their corresponding side lengths.

The factors of

36

greater than

7

are

9, 12, 18,

and

36

. This yields four potential cases:
- If

7+x = 9 \implies x = 2

, then

k = 4

, and the sides are

12, 16, 8

.
- If

7+x = 12 \implies x = 5

, then

k = 3

, and the sides are

9, 12, 15

.
- If

7+x = 18 \implies x = 11

, then

k = 2

, and the sides are

6, 8, 22

.
- If

7+x = 36 \implies x = 29

, then

k = 1

, and the sides are

3, 4, 29

.

This identifies all mathematically possible configurations before checking if they can physically form a triangle.

4

Apply the Triangle Inequality Theorem to each case.

The sum of the lengths of any two sides must be strictly greater than the length of the remaining side:
- For sides

12, 16, 8

:

12 + 8 = 20 > 16

(Valid).
- For sides

9, 12, 15

:

9 + 12 = 21 > 15

(Valid).
- For sides

6, 8, 22

:

6 + 8 = 14 < 22

(Invalid).
- For sides

3, 4, 29

:

3 + 4 = 7 < 29

(Invalid).
Only the cases where

x = 2

and

x = 5

form valid triangles.

This filters the candidate values of

x

to only those that can form a valid geometric triangle.

Key Concept

Triangle Inequality Theorem and Integer Ratio Constraints

Estimated Time:1m 30s

Question 53Question →

A convex pentagon has four interior angles that measure $80^\circ$ , $110^\circ$ , $120^\circ$ , and $130^\circ$ . What is the degree measure of the fifth interior angle?

Show answer & explanation

Answer: 100

Answer

The degree measure of the fifth interior angle is

100^\circ

.

The sum of the interior angles of a pentagon (

n = 5

) is

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

. The sum of the four given angles is

80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ

. The measure of the fifth angle is the difference between these two values:

540^\circ - 440^\circ = 100^\circ

.

Step-by-Step Solution

1

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

The sum of the interior angles is

540^\circ

.

The sum of the interior angles of any

n

-sided convex polygon is calculated using the formula

(n - 2) \times 180^\circ

. For a pentagon,

n = 5

, which gives

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

.

2

Sum the measures of the four given interior angles.

The sum of the four given angles is

440^\circ

.

Adding the given measures:

80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ

.

3

Subtract the sum of the four given angles from the total sum of the interior angles.

The measure of the fifth interior angle is

100^\circ

.

Subtracting the sum of the known angles from the total pentagon interior angle sum yields

540^\circ - 440^\circ = 100^\circ

.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

.

Question 54Question →

A right triangle has legs of length $5$ inches and $12$ inches. What is the length, in inches, of the hypotenuse of this triangle?

Show answer & explanation

Answer: 13

Answer

The length of the hypotenuse is 13 inches.

The Pythagorean theorem states that in any right triangle with legs

a

and

b

and hypotenuse

c

,

a^2 + b^2 = c^2

. Substituting the given values:

5^2 + 12^2 = 25 + 144 = 169

. Taking the square root of

169

gives the hypotenuse length of

13

.

Step-by-Step Solution

1

Identify the lengths of the two legs.

a = 5

,

b = 12

These are the given side lengths perpendicular to each other.

2

Apply the Pythagorean theorem

a^2 + b^2 = c^2

.

5^2 + 12^2 = 25 + 144 = 169

To find the square of the hypotenuse.

3

Solve for the hypotenuse

c

by taking the square root.

c = \sqrt{169} = 13

To find the side length of the hypotenuse.

Key Concept

Pythagorean Theorem

Question 55Question →

A $13$ -foot ladder is leaning against a flat vertical wall. The base of the ladder is placed $5$ feet away from the bottom of the wall. How many feet up the wall does the ladder reach?

\sqrt{8}

8

12

\sqrt{194}

18

Show answer & explanation

Answer: $12$

Answer

The height the ladder reaches is

12

feet.

The correct answer is

12

feet. The ladder, wall, and ground form a right triangle where the ladder is the hypotenuse (

13

feet) and the distance along the ground is one leg (

5

feet). By the Pythagorean theorem, the height up the wall,

b

, satisfies

5^2 + b^2 = 13^2

. Solving for

b

gives

b^2 = 169 - 25 = 144

, so

b = 12

.

Step-by-Step Solution

1

Identify the hypotenuse and the given leg from the word problem description.

The ladder length is the hypotenuse (

c = 13

), and the distance from the wall is one of the legs (

a = 5

).

The ladder forms the diagonal side opposite the right angle formed by the vertical wall and the ground.

2

Set up the Pythagorean theorem to find the unknown leg length.

5^2 + b^2 = 13^2

.

The Pythagorean theorem (

a^2 + b^2 = c^2

) relates the side lengths of any right triangle.

3

Solve for the unknown leg length

b

by simplifying terms and taking the square root.

25 + b^2 = 169 \implies b^2 = 144 \implies b = \sqrt{144} = 12

.

Subtracting the square of the known leg from the square of the hypotenuse isolates the squared unknown leg, which can then be solved by finding its square root.

Key Concept

Using the Pythagorean theorem to find an unknown leg of a right triangle when the hypotenuse and one leg are known.

Alternative Method

Recognizing that

5

and

13

are part of the common Pythagorean triple

5

-

12

-

13

allows you to immediately identify the missing leg as

12

without performing calculations.

Estimated Time:45s

Question 56Question →

In the standard $(x,y)$ coordinate plane, a line segment $OQ$ connects the origin $O(0,0)$ to a point $Q$ in the first quadrant. The segment $OQ$ has a length of $10$ units and makes an angle of $60^\circ$ with the positive $x$ -axis. An isosceles right triangle $\triangle OQR$ is constructed such that the right angle is at $Q$ , the leg $QR$ has a length of $10$ units, and point $R$ lies in the first quadrant. What are the coordinates of point $R$ ?

(5\sqrt{3} - 5, 5\sqrt{3} + 5)

(5\sqrt{3} + 5, 5 - 5\sqrt{3})

(5 + 5\sqrt{3}, 5\sqrt{3} - 5)

(20 - 10\sqrt{3}, 10)

(10 + 10\sqrt{3}, 10\sqrt{3} - 10)

Show answer & explanation

Answer: $(5 + 5\sqrt{3}, 5\sqrt{3} - 5)$

Answer

The coordinates

(5 + 5\sqrt{3}, 5\sqrt{3} - 5)

The correct answer is

(5 + 5\sqrt{3}, 5\sqrt{3} - 5)

because constructing two helper

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangles allows us to determine both the coordinates of

Q

as

(5, 5\sqrt{3})

and the horizontal and vertical shifts to

R

as

+5\sqrt{3}

and

-5

respectively.

Step-by-Step Solution

1

Project point

Q

onto the

x

-axis to form a

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangle.

The horizontal leg is

10 \cos(60^\circ) = 5

and the vertical leg is

10 \sin(60^\circ) = 5\sqrt{3}

.

The hypotenuse

OQ

has a length of

10

and makes a

60^\circ

angle with the positive

x

-axis.

2

Determine the coordinates of point

Q

.

Q = (5, 5\sqrt{3})

.

Point

Q

is in the first quadrant, so both coordinates are positive.

3

Determine the orientation of segment

QR

.

Segment

QR

must make a

30^\circ

angle below the horizontal line passing through

Q

(going down and to the right).

Since

\triangle OQR

is a right isosceles triangle with the right angle at

Q

,

QR

is perpendicular to

OQ

and has length

10

. To keep

R

in the first quadrant,

QR

must rotate clockwise from

OQ

by

90^\circ

.

4

Construct a helper

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangle under

QR

to find the changes in

x

and

y

.

The horizontal change is

+10 \cos(30^\circ) = +5\sqrt{3}

and the vertical change is

-10 \sin(30^\circ) = -5

.

The hypotenuse of this triangle is

QR = 10

, and the angle with the horizontal is

30^\circ

.

5

Calculate the coordinates of

R

by applying the changes to the coordinates of

Q

.

R = (5 + 5\sqrt{3}, 5\sqrt{3} - 5)

.

Add the horizontal change to

x_Q

and the vertical change to

y_Q

.

Key Concept

Solving coordinate geometry problems using

30^\circ\text{-}60^\circ\text{-}90^\circ

special right triangles.

Question 57Question →

A regular octagon has 8 congruent interior angles. What is the degree measure of one of these interior angles?

Show answer & explanation

Answer: 135

Answer

The degree measure of one interior angle of a regular octagon is 135 degrees.

The interior angles of a regular octagon sum to

1080^\circ

because

(8-2) \times 180^\circ = 1080^\circ

. Since a regular octagon has 8 equal angles, dividing

1080^\circ

by

8

results in

135^\circ

per interior angle.

Step-by-Step Solution

1

Identify the number of sides in a regular octagon.

The number of sides,

n

, is

8

.

An octagon is defined as a polygon with 8 sides and 8 angles.

2

Calculate the sum of the interior angles of the octagon.

The sum of the interior angles is

1080^\circ

.

The sum of the interior angles of any convex polygon with

n

sides is

(n-2) \times 180^\circ

. For

n = 8

, this is

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

.

3

Find the measure of one interior angle by dividing the sum by the number of angles.

Each interior angle measures

135^\circ

.

A regular polygon has congruent interior angles. Therefore, dividing the total sum of

1080^\circ

by the

8

congruent angles yields the measure of each individual interior angle:

1080^\circ \div 8 = 135^\circ

.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

. For a regular polygon, each interior angle has a measure of

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

.

Question 58Question →

A triangle has side lengths of $7$ , $x+2$ , and $2x-1$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

Show answer & explanation

Answer: 7

Answer

The total number of possible integer values for

x

is

7

.

According to the Triangle Inequality Theorem, a triangle is formed if and only if the sum of any two side lengths is strictly greater than the third side length. Solving the three inequalities

7 + (x+2) > 2x-1

,

7 + (2x-1) > x+2

, and

(x+2) + (2x-1) > 7

gives

x < 10

,

x > -4

, and

x > 2

. The overlapping interval is

2 < x < 10

. The integers in this range are

\{3, 4, 5, 6, 7, 8, 9\}

, which gives a total of

7

possible integer values.

Step-by-Step Solution

1

Set up the three inequalities required by the Triangle Inequality Theorem.

1)

7 + (x+2) > 2x-1

2)

7 + (2x-1) > x+2

3)

(x+2) + (2x-1) > 7

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side.

2

Solve the first inequality:

7 + (x+2) > 2x-1

.

x < 10

Simplifying the left side yields

x + 9 > 2x - 1

. Subtracting

x

and adding

1

to both sides results in

10 > x

, which means

x < 10

.

3

Solve the second inequality:

7 + (2x-1) > x+2

.

x > -4

Simplifying the left side yields

2x + 6 > x + 2

. Subtracting

x

and

6

from both sides results in

x > -4

.

4

Solve the third inequality:

(x+2) + (2x-1) > 7

.

x > 2

Simplifying the left side yields

3x + 1 > 7

. Subtracting

1

and dividing by

3

results in

x > 2

.

5

Determine the combined range for

x

and identify the valid integers.

The combined range is

2 < x < 10

. The valid integers are

3, 4, 5, 6, 7, 8,

and

9

.

To satisfy all three inequalities,

x

must be greater than

2

, greater than

-4

, and less than

10

, which simplifies to

2 < x < 10

.

6

Count the total number of valid integer values for

x

.

There are

7

integer values.

Counting the elements of the set

\{3, 4, 5, 6, 7, 8, 9\}

yields a total of

7

values.

Key Concept

Triangle Inequality Theorem

Estimated Time:1m 30s

Question 59Question →

For a certain four-sided convex polygon, the ratio of its interior angle measures is $1:2:3:4$ . What is the degree measure of the smallest angle?

18^\circ

54^\circ

36^\circ

72^\circ

90^\circ

Show answer & explanation

Answer: $36^\circ$

Answer

36^\circ

The correct answer is

36^\circ

because the sum of the interior angles of a four-sided polygon is

360^\circ

. The ratio

1:2:3:4

means the angles can be represented as

x

,

2x

,

3x

, and

4x

. Their sum is

10x = 360^\circ

, which yields

x = 36^\circ

. The smallest angle is

x

, which is

36^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles of a four-sided convex polygon.

The sum of the interior angles of a four-sided polygon is

360^\circ

.

The sum of the interior angles of a polygon with

n

sides is given by the formula

(n-2) \times 180^\circ

. For a four-sided polygon,

(4-2) \times 180^\circ = 360^\circ

.

2

Find the sum of the parts of the given ratio.

1 + 2 + 3 + 4 = 10

parts

To distribute the total angle measure proportionally, the individual parts of the ratio

1:2:3:4

must be summed.

3

Calculate the measure of one part of the ratio.

360^\circ \div 10 = 36^\circ

Dividing the total sum of the interior angles by the total number of parts determines the degree measure of a single part.

4

Find the measure of the smallest angle.

1 \times 36^\circ = 36^\circ

The smallest angle corresponds to the smallest part of the ratio, which is 1.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and this total can be divided proportionally using a given ratio to find individual angle measures.

Estimated Time:1m 0s

Question 60Question →

A rectangular piece of paper $ABCD$ has dimensions $AB = 12$ inches and $BC = 9$ inches. The paper is folded so that vertex $A$ falls directly on vertex $C$ , creating a crease $EF$ where $E$ lies on $AB$ and $F$ lies on $CD$ . What is the length of the crease $EF$ , in inches?

Show answer & explanation

Answer: 11.25

Answer

The length of the crease is 11.25 inches.

By interpreting the fold geometrically, we establish that the crease is the perpendicular bisector of the diagonal. We can solve for the segments along the side using a right triangle formed at the corner, and then construct a second right triangle using an altitude to find the length of the crease hypotenuse.

Step-by-Step Solution

1

Identify the relationship created by the fold.

The crease

EF

is the perpendicular bisector of the diagonal

AC

, meaning

AE = EC = x

.

When a point is folded onto another, the crease line acts as the perpendicular bisector of the segment connecting the two points.

2

Set up an expression for the remaining part of the side

AB

.

Since

AB = 12

and

AE = x

, the length of

EB = 12 - x

.

The point

E

lies on segment

AB

, dividing it into

AE

and

EB

.

3

Use the Pythagorean Theorem in right triangle

EBC

to solve for

x

.

x^2 = (12 - x)^2 + 9^2 \implies x^2 = 144 - 24x + x^2 + 81 \implies 24x = 225 \implies x = 9.375

.

The triangle

EBC

is a right triangle with legs

EB

and

BC

, and hypotenuse

EC

.

4

Form a second right triangle to find the length of the crease

EF

.

Draw

FG \perp AB

with

G

on

AB

. This forms right triangle

EGF

with legs

FG = 9

and

EG = AB - EB - DF = 12 - 2.625 - 2.625 = 6.75

.

Constructing an altitude from

F

to

AB

allows us to create a right triangle that has the crease

EF

as its hypotenuse.

5

Apply the Pythagorean Theorem to right triangle

EGF

to calculate the final length of

EF

.

EF = \sqrt{6.75^2 + 9^2} = \sqrt{45.5625 + 81} = \sqrt{126.5625} = 11.25

inches.

The hypotenuse of right triangle

EGF

represents the length of the crease.

Key Concept

Applying the Pythagorean Theorem to geometric folds and multi-step right triangle relationships

Plane Geometry

Answer

Step-by-Step Solution

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Common Mistakes

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Alternative Method

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