Plane Geometry Practice Questions — ACT — Page 2

Question 21Question →

A triangle has two sides of lengths $5$ centimeters and $11$ centimeters. Which of the following could be the perimeter of the triangle, in centimeters?

16

22

27

32

34

Show answer & explanation

Answer: 27

Answer

27

The correct answer is

27

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

x

, of a triangle must be strictly greater than the difference of the other two sides (

11 - 5 = 6

) and strictly less than their sum (

11 + 5 = 16

). This gives the inequality range

6 < x < 16

. The perimeter is the sum of all three sides, which is

P = 5 + 11 + x = 16 + x

. Applying the bounds of

x

, we find that the perimeter must satisfy

16 + 6 < P < 16 + 16

, which simplifies to

22 < P < 32

. Among the options,

27

is the only value that is strictly within this range.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to find the limits for the third side.

Let the third side be

x

. The length of

x

must satisfy:

11 - 5 < x < 11 + 5 \implies 6 < x < 16

.

The length of any side of a triangle must be strictly between the positive difference and the sum of the lengths of the other two sides.

2

Set up the equation for the perimeter of the triangle.

Perimeter

P = 5 + 11 + x = 16 + x

.

The perimeter of a triangle is defined as the sum of the lengths of all three of its sides.

3

Find the range of possible values for the perimeter

P

by applying the inequality bounds of

x

.

Add

16

to all parts of the inequality

6 < x < 16

:

16 + 6 < 16 + x < 16 + 16 \implies 22 < P < 32

.

We must shift the inequality bounds of the third side by the sum of the two known sides to find the range of the perimeter.

4

Identify the option that falls strictly inside the range

22 < P < 32

.

The value

27

is the only option that satisfies

22 < 27 < 32

.

Only a value strictly between

22

and

32

can represent a mathematically valid perimeter for this triangle.

Key Concept

Triangle Inequality Theorem and Perimeter

Estimated Time:1m 0s

Question 22Question →

A triangle has two sides of length $8$ and $15$ . If the length of the third side, $s$ , is a prime number, how many possible values are there for $s$ ?

3

4

5

6

7

Show answer & explanation

Answer: $4$

Answer

There are 4 possible values for the third side length s.

The correct answer is

4

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

s

of a triangle with sides of

8

and

15

must satisfy

15 - 8 < s < 15 + 8

. This simplifies to the open interval

7 < s < 23

. The prime numbers strictly between

7

and

23

are

11

,

13

,

17

, and

19

. Counting these gives exactly

4

possible prime values for

s

.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to determine the bounds for the third side

s

.

15 - 8 < s < 15 + 8

, which simplifies to

7 < s < 23

.

The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.

2

Identify all prime numbers that lie strictly within the range

(7, 23)

.

The prime numbers in this range are

11

,

13

,

17

, and

19

.

A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself.

3

Count the number of identified prime numbers.

There are

4

prime numbers (

11

,

13

,

17

,

19

).

This counts the total number of possible valid lengths for

s

.

Key Concept

Triangle Inequality Theorem and basic number properties

Question 23Question →

In a triangle, two of the sides have lengths $13$ and $20$ . The third side has a length of $s$ , where $s$ is an integer. If the side of length $20$ is the longest side of the triangle, and the triangle is obtuse, what is the number of possible values for $s$ ?

Show answer & explanation

Answer: 8

Answer

There are 8 possible integer values for

s

.

To find the number of possible integer values for

s

, we combine the Triangle Inequality Theorem (

13 + s > 20 \implies s > 7

) and the condition for an obtuse triangle with

20

as the longest side (

20^2 > 13^2 + s^2 \implies s^2 < 231 \implies s \leq 15

). This limits

s

to integers in the range

[8, 15]

, which contains exactly

8

values.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to find the lower bound for

s

.

s > 7

, so the minimum integer value is

8

.

The sum of the two shorter sides of a triangle must be strictly greater than the longest side.

2

Set up the obtuse triangle inequality with

20

as the longest side.

20^2 > 13^2 + s^2

In any obtuse triangle with longest side

c

, the inequality

c^2 > a^2 + b^2

must hold.

3

Solve the inequality

20^2 > 13^2 + s^2

for

s

.

s^2 < 231 \implies s \leq 15

Simplifying the inequality gives

400 > 169 + s^2 \implies s^2 < 231

. The largest integer whose square is less than

231

is

15

.

4

Determine the number of integers in the range

[8, 15]

.

8 possible values

The integers satisfying both conditions are

\{8, 9, 10, 11, 12, 13, 14, 15\}

, which count to

8

.

Key Concept

Triangle Inequality Theorem and obtuse triangle classification using side lengths

Alternative Method

List the perfect squares and verify which ones satisfy both

s^2 < 231

and the Triangle Inequality Theorem

s > 7

.

Estimated Time:2m 0s

Question 24Question →

A non-degenerate triangle has side lengths of $5$ , $12$ , and $x$ . A second non-degenerate triangle has side lengths of $x$ , $10$ , and $y$ . If $x$ and $y$ must be integers, and the perimeter of the second triangle is the minimum possible integer value, what is the sum of all possible values of $y$ ?

1

3

6

10

27

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By applying the Triangle Inequality Theorem, the shared side length

x

of the first triangle must satisfy

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

. Since

x

must be an integer, its possible values are

\{8, 9, 10, 11, 12, 13, 14, 15, 16\}

. For the second triangle with side lengths

x

,

10

, and

y

, the Triangle Inequality Theorem requires

|x - 10| < y < x + 10

. To minimize the perimeter

P = x + 10 + y

, we minimize

x + y

. Checking the possible values of

x

, we find that when

x = 8

, the minimum integer value for

y

is

3

(giving

P = 21

); when

x = 9

, the minimum integer value for

y

is

2

(giving

P = 21

); and when

x = 10

, the minimum integer value for

y

is

1

(giving

P = 21

). For any

x \ge 11

, the minimum value of

y

is

x - 9

, resulting in a perimeter of at least

23

. Therefore, the minimum perimeter of the second triangle is

21

, which is achieved when

y

is

3

,

2

, or

1

. The sum of these values of

y

is

1 + 2 + 3 = 6

.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to the first triangle to find the range of the shared side length

x

.

7 < x < 17

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

12 - 5 < x < 12 + 5

.

2

List the possible integer values of

x

based on the range.

x \in \{8, 9, 10, 11, 12, 13, 14, 15, 16\}

The problem states that

x

must be an integer.

3

Apply the Triangle Inequality Theorem to the second triangle with sides

x

,

10

, and

y

to express the range of

y

in terms of

x

.

|x - 10| < y < x + 10

The third side length

y

must be strictly between the difference and the sum of the other two sides (

x

and

10

).

4

Determine the minimum integer value of the perimeter of the second triangle,

P = x + 10 + y

, by testing the possible values of

x

and finding the minimum integer

y

for each.

The minimum perimeter is

21

, achieved when

(x, y) = (8, 3)

,

(9, 2)

, or

(10, 1)

.

Minimizing the perimeter

P = x + y + 10

is equivalent to minimizing the sum

x + y

for integer values of

y > |x - 10|

.

5

Sum the possible integer values of

y

that yield the minimum perimeter.

3 + 2 + 1 = 6

We need to find the sum of all possible values of

y

that result in the minimum perimeter of

21

.

Key Concept

Triangle Inequality Theorem and Perimeter Optimization

Estimated Time:3m 0s

Question 25Question →

The measures of the interior angles of a triangle are in the ratio $2:3:7$ . If the measure of the largest angle is decreased by $15^\circ$ and the measure of the smallest angle is increased by $15^\circ$ , what is the ratio of the interior angles of the new triangle, ordered from smallest to largest?

1:2:3

1:3:8

1:1:2

2:3:6

5:6:13

Show answer & explanation

Answer: $1:1:2$

Answer

The ratio of the interior angles of the new triangle is

1:1:2

.

The correct answer is the ratio

1:1:2

. The sum of the interior angles in a triangle is always

180^\circ

. Given the ratio

2:3:7

, the sum of the parts is

12

, which means each part represents

180^\circ / 12 = 15^\circ

. The original angles are therefore

30^\circ

,

45^\circ

, and

105^\circ

. Increasing the smallest angle by

15^\circ

gives

45^\circ

, and decreasing the largest by

15^\circ

gives

90^\circ

. The new angle measures are

45^\circ

,

45^\circ

, and

90^\circ

, which simplifies to

1:1:2

.

Step-by-Step Solution

1

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

2x + 3x + 7x = 180^\circ

The sum of the interior angles of any triangle is always

180^\circ

.

2

Solve for the value of

x

.

12x = 180^\circ \implies x = 15^\circ

Combining like terms simplifies the equation to find the value of one ratio unit.

3

Calculate the original measures of the three angles.

Smallest:

30^\circ

, Middle:

45^\circ

, Largest:

105^\circ

Multiply each part of the ratio by

x = 15^\circ

.

4

Apply the modifications to the smallest and largest angles.

New smallest:

30^\circ + 15^\circ = 45^\circ

; New largest:

105^\circ - 15^\circ = 90^\circ

; Middle:

45^\circ

(unchanged).

Perform the operations described in the problem statement.

5

Order the new angle measures from smallest to largest and simplify the ratio.

45^\circ : 45^\circ : 90^\circ \implies 1 : 1 : 2

Divide each term in the ratio by the greatest common divisor, which is

45

.

Key Concept

Angle sum theorem of a triangle and ratio partition applications

Alternative Method

Instead of calculating the actual angle values, note that the sum of the ratio parts is

2+3+7 = 12

, and the sum of the interior angles of a triangle is

180^\circ

. This means

1

ratio unit is equal to

180^\circ / 12 = 15^\circ

. Since the smallest angle is increased by

15^\circ

(exactly

1

ratio unit) and the largest is decreased by

15^\circ

(exactly

1

ratio unit), we can apply these modifications directly to the ratio terms. The new ratio terms are

2+1 = 3

,

3

(unchanged), and

7-1 = 6

. This gives a ratio of

3:3:6

, which simplifies to

1:1:2

.

Estimated Time:1m 30s

Question 26Question →

In $\triangle ABC$ , the side lengths are $AB = 12$ , $BC = 15$ , and $AC = 18$ . A point $P$ lies strictly inside $\triangle ABC$ . If the lengths of the segments $BP$ and $CP$ are both integers, what is the maximum possible value of the sum of these two lengths?

26

27

28

29

30

Show answer & explanation

Answer: 28

Answer

28

The correct answer is 28. According to the properties of triangles, for any point

P

strictly inside

\triangle ABC

, the sum of the interior segments is strictly less than the sum of the other two sides:

BP + CP < AB + AC = 12 + 18 = 30

. Since

BP

and

CP

are integers, we check the boundary where

P

lies on the side

AC

at an integer distance

CP = z

from

C

. Applying Stewart's Theorem, the boundary length

BP = y_{bound}

is

\sqrt{z^2 - 22.5z + 225}

. For the largest possible integer value

z = 17

, the boundary value is approximately

11.47

. Because the point must lie strictly inside the triangle,

BP

must be strictly less than this boundary, so the maximum integer value for

BP

is 11. This yields a maximum sum of

11 + 17 = 28

. Lower integer values of

z

yield smaller maximum sums (for example, if

z = 16

, the boundary is exactly 11, so

BP

can be at most 10, giving a sum of 26).

Step-by-Step Solution

1

Apply the interior point triangle inequality theorem.

For any point

P

strictly inside

\triangle ABC

, the sum of the distances to two vertices is strictly less than the sum of the other two sides:

BP + CP < AB + AC

.

This establishes the theoretical upper bound for the sum of the two segment lengths.

2

Calculate the theoretical upper bound.

Since

AB = 12

and

AC = 18

, we have

BP + CP < 12 + 18 = 30

. Since

BP

and

CP

must be integers, the sum

BP + CP

can be at most 29.

This sets the initial integer limit before evaluating if it is geometrically possible.

3

Analyze the boundary conditions for integer lengths using Stewart's Theorem.

Let

CP = z

and

BP = y

, where

y

and

z

are integers. As

P

approaches the side

AC

, the boundary value

y_{bound}

represents the distance from

B

to a point on

AC

at distance

z

from

C

. Using Stewart's Theorem, this boundary satisfies:

y_{bound}^2 = z^2 - 22.5z + 225

. Since

P

is strictly inside the triangle,

y

must be strictly less than

y_{bound}

.

This provides the mathematical relationship determining whether a point is inside the triangle for any given integer length of one segment.

4

Test the maximum possible integer value for

z

to maximize the sum

y + z

.

Since

P

is strictly inside,

z

must be strictly less than

AC = 18

, so the maximum integer for

z

is 17. For

z = 17

, the boundary value is

y_{bound} = \sqrt{17^2 - 22.5(17) + 225} = \sqrt{131.5} \approx 11.47

. Since

y < y_{bound}

, the maximum integer value for

y

is 11. This yields a maximum sum of

11 + 17 = 28

.

This determines the actual maximum integer sum that can be geometrically realized within the triangle.

Key Concept

Triangle Inequality Theorem and Interior Point Properties

Question 27Question →

A square tabletop has a diagonal length of $8$ feet. What is the area of the tabletop, in square feet?

16

16\sqrt{2}

32

32\sqrt{2}

64

Show answer & explanation

Answer: 32

Answer

32

The correct answer is 32. The diagonal of a square divides it into two congruent 45-45-90 right triangles, where the diagonal is the hypotenuse. The ratio of the diagonal to the side length in a 45-45-90 triangle is

\sqrt{2}

to

1

. Therefore, a square with a diagonal of

8

feet has a side length of

s = \frac{8}{\sqrt{2}}

feet. The area of the square is

s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 = \frac{64}{2} = 32

square feet.

Step-by-Step Solution

1

Relate the diagonal of a square to its side length using special right triangles.

The diagonal of a square splits the square into two 45-45-90 right triangles. The hypotenuse of these triangles is the diagonal,

8

feet, and the legs are the sides of the square,

s

. The relationship is

s\sqrt{2} = 8

.

In a 45-45-90 triangle, the hypotenuse is

\sqrt{2}

times the length of a leg.

2

Solve for the side length

s

of the square tabletop.

s = \frac{8}{\sqrt{2}}

feet.

Divide both sides of the equation by

\sqrt{2}

to isolate the side length

s

.

3

Calculate the area of the square tabletop.

Area

= s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 = \frac{64}{2} = 32

square feet.

The area of a square is calculated by squaring its side length.

Key Concept

Using the properties of 45-45-90 special right triangles to find side lengths and area from the diagonal of a square.

Question 28Question →

In $\triangle ABC$ , the lengths of sides $AB$ and $AC$ are both $13$ . A point $D$ lies on side $BC$ such that $AD$ is an integer. If the perimeter of $\triangle ABD$ is equal to the perimeter of $\triangle ACD$ , what is the sum of all possible integer values for the length of $BC$ ?

Show answer & explanation

Answer: 34

Answer

34

The perimeters of

\triangle ABD

and

\triangle ACD

are equal, and since

AB = AC = 13

, this forces

BD = CD

, making

D

the midpoint of

BC

. In the isosceles triangle

\triangle ABC

, the median

AD

is perpendicular to

BC

, making

\triangle ABD

a right triangle. By the Pythagorean theorem,

BD^2 + AD^2 = 169

. Since

AD

is an integer,

BD

must also be an integer (a half-integer would result in

AD^2

ending in

.25

, which cannot be a perfect square of an integer). The only positive integer solutions for

(BD, AD)

are

(5, 12)

and

(12, 5)

. This results in

BC = 2 \cdot BD

being either

10

or

24

. The sum of these possible values is

10 + 24 = 34

.

Step-by-Step Solution

1

Set the perimeters of

\triangle ABD

and

\triangle ACD

equal to each other.

BD = CD

Since

AB = AC = 13

, equating

AB + BD + AD = AC + CD + AD

simplifies directly to

BD = CD

.

2

Determine the relationship between

AD

and

BC

.

\triangle ABD

is a right triangle with hypotenuse

13

.

In an isosceles triangle, the median to the base is also the altitude, so

AD \perp BC

.

3

Apply the Pythagorean theorem to

\triangle ABD

.

BD^2 + AD^2 = 169

The sum of the squares of the legs in right triangle

\triangle ABD

must equal the square of the hypotenuse

AB = 13

.

4

Analyze the parity and integer constraints of

BD

and

AD

.

BD

must be a positive integer.

If

BD

were a half-integer,

BD^2

would end in

.25

, preventing

AD^2

from being an integer, which contradicts the given condition that

AD

is an integer.

5

Identify the Pythagorean triples with a hypotenuse of

13

.

(BD, AD) \in \{(5, 12), (12, 5)\}

The only positive integer solutions to

x^2 + y^2 = 13^2

are

(5, 12)

and

(12, 5)

.

6

Calculate the possible lengths of

BC

and sum them.

BC \in \{10, 24\}

, and their sum is

34

.

Since

D

is the midpoint of

BC

, the length of

BC

is

2 \cdot BD

, yielding

2 \cdot 5 = 10

and

2 \cdot 12 = 24

. Both satisfy the triangle inequality because

BC < AB + AC = 26

.

Key Concept

Properties of Isosceles Triangles and the Pythagorean Theorem

Estimated Time:3m 0s

Question 29Question →

A right triangle has two legs of equal length. If the hypotenuse of the triangle is $10\sqrt{2}$ centimeters, what is the length, in centimeters, of one of the legs?

Show answer & explanation

Answer: 10

Answer

The length of one of the legs is 10 centimeters.

An isosceles right triangle possesses acute angles of

45^\circ

and side ratios of

x : x : x\sqrt{2}

, where

x

represents the leg length. Given a hypotenuse of

10\sqrt{2}

centimeters, we equate

x\sqrt{2} = 10\sqrt{2}

. Dividing both sides of the equation by

\sqrt{2}

isolates the leg length, giving

x = 10

centimeters.

Step-by-Step Solution

1

Determine the triangle type from the given properties.

The triangle is a

45^\circ

-

45^\circ

-

90^\circ

special right triangle (isosceles right triangle).

A right triangle with two legs of equal length must have acute angles measuring

45^\circ

each, making it an isosceles right triangle.

2

Set up an equation utilizing the ratios of the side lengths.

Let

x

be the leg length. The hypotenuse length is represented by

x\sqrt{2} = 10\sqrt{2}

centimeters.

The hypotenuse of a

45^\circ

-

45^\circ

-

90^\circ

special right triangle is always

\sqrt{2}

times the length of one of its legs.

3

Solve the equation for the variable

x

.

x = 10

Dividing both sides of the equation by

\sqrt{2}

isolates the variable

x

representing the leg length.

Key Concept

Properties of

45^\circ

-

45^\circ

-

90^\circ

special right triangles.

Alternative Method

Alternatively, you can apply the Pythagorean Theorem:

a^2 + b^2 = c^2

. Since both legs are equal in length, we can set

a = b = x

. This yields the equation

x^2 + x^2 = (10\sqrt{2})^2

. Simplifying both sides gives

2x^2 = 100 \times 2 = 200

. Dividing by 2 yields

x^2 = 100

, and taking the square root of both sides gives

x = 10

centimeters.

Estimated Time:45s

Question 30Question →

In $\triangle ABC$ , the measure of $\angle B$ is $80^\circ$ and the measure of $\angle C$ is $40^\circ$ . A point $D$ lies on side $BC$ such that $AD$ bisects $\angle BAC$ , and a point $E$ lies on side $AC$ such that $AD = AE$ . What is the measure, in degrees, of $\angle CDE$ ?

Show answer & explanation

Answer: 35

Answer

35

The correct answer is

35

. By first finding that

\angle BAC = 60^\circ

, we use the angle bisector

AD

to find

\angle CAD = 30^\circ

. In

\triangle ADC

, we find the interior angle

\angle ADC = 110^\circ

. In the isosceles triangle

\triangle ADE

with

AD=AE

, the base angles are

\angle ADE = \angle AED = 75^\circ

. Finally, subtracting

\angle ADE

from

\angle ADC

gives

\angle CDE = 35^\circ

.

Step-by-Step Solution

1

Find the measure of the third angle of the main triangle,

\angle BAC

.

\angle BAC = 60^\circ

The sum of the interior angles of any triangle is

180^\circ

. Therefore,

\angle BAC = 180^\circ - \angle B - \angle C = 180^\circ - 80^\circ - 40^\circ = 60^\circ

.

2

Determine the measure of the bisected angle

\angle CAD

.

\angle CAD = 30^\circ

Since

AD

bisects

\angle BAC

, it divides the angle into two equal parts:

\angle BAD = \angle CAD = \frac{60^\circ}{2} = 30^\circ

.

3

Calculate the interior angle

\angle ADC

in

\triangle ADC

.

\angle ADC = 110^\circ

In

\triangle ADC

, the sum of angles is

180^\circ

. Therefore,

\angle ADC = 180^\circ - \angle CAD - \angle C = 180^\circ - 30^\circ - 40^\circ = 110^\circ

.

4

Find the base angles of the isosceles triangle

ADE

.

\angle ADE = 75^\circ

Since

AD = AE

,

\triangle ADE

is an isosceles triangle with vertex angle

\angle DAE = 30^\circ

. The two base angles,

\angle ADE

and

\angle AED

, are equal. Thus,

\angle ADE = \frac{180^\circ - 30^\circ}{2} = 75^\circ

.

5

Determine the final angle

\angle CDE

by subtraction.

\angle CDE = 35^\circ

Since point

E

lies on side

AC

, ray

DE

lies between rays

DA

and

DC

. Therefore,

\angle ADC = \angle ADE + \angle CDE

. Rearranging gives

\angle CDE = \angle ADC - \angle ADE = 110^\circ - 75^\circ = 35^\circ

.

Key Concept

Applying triangle angle sum theorem, angle bisector properties, and isosceles triangle base angle properties to perform multi-step angle tracing.

Alternative Method

Use the exterior angle theorem on

\triangle ADC

at vertex

D

:

\angle ADB = \angle CAD + \angle C = 30^\circ + 40^\circ = 70^\circ

. Then, since

E

is on

AC

,

A

,

E

, and

C

are collinear. In

\triangle ADE

, the exterior angle at

E

is

\angle DEC = \angle DAE + \angle ADE = 30^\circ + 75^\circ = 105^\circ

. In

\triangle DEC

, the sum of angles is

180^\circ

, so

\angle CDE = 180^\circ - 105^\circ - 40^\circ = 35^\circ

.

Estimated Time:2m 30s

Question 31Question →

In a certain triangle, the ratio of the measure of the first angle to the measure of the second angle is $1:2$ . The measure of the third angle is $20^\circ$ less than the measure of the second angle. What is the measure of the largest angle in the triangle?

40^\circ

64^\circ

80^\circ

100^\circ

120^\circ

Show answer & explanation

Answer: $80^\circ$

Answer

80^\circ

The correct answer is

80^\circ

. By representing the first angle as

x

, the second angle is

2x

, and the third angle is

2x - 20

. The sum of the interior angles in a triangle is always

180^\circ

, which gives the equation

x + 2x + (2x - 20) = 180

. Simplifying this results in

5x - 20 = 180

, which yields

5x = 200

and

x = 40

. Substituting this value back into the expressions for the three angles gives measures of

40^\circ

,

80^\circ

, and

60^\circ

. Comparing these values shows that the largest angle is

80^\circ

.

Step-by-Step Solution

1

Define the measures of the first and second angles using a single variable based on their ratio.

Let the first angle be

x

and the second angle be

2x

.

Since the ratio of the first angle to the second angle is

1:2

, we can represent them as

x

and

2x

respectively.

2

Express the measure of the third angle in terms of the same variable.

The third angle is

2x - 20

.

The problem states the third angle is

20^\circ

less than the second angle, which has a measure of

2x

.

3

Set up an equation using the triangle angle sum theorem and solve for

x

.

x + 2x + (2x - 20) = 180 \implies 5x - 20 = 180 \implies 5x = 200 \implies x = 40

.

The sum of the measures of the interior angles of any triangle is always

180^\circ

.

4

Calculate the measures of all three angles to determine which is the largest.

First angle =

40^\circ

, Second angle =

2(40^\circ) = 80^\circ

, Third angle =

2(40^\circ) - 20^\circ = 60^\circ

. The largest angle is

80^\circ

.

We must substitute

x = 40

back into our expressions for each angle to find their actual degree measures and identify the largest one.

Key Concept

The interior angles of a triangle always sum to

180^\circ

. Ratios and word problems can be modeled algebraically to determine unknown angle measures.

Estimated Time:1m 15s

Question 32Question →

In $\triangle ABC$ , point $D$ lies on side $BC$ . The segment $AD$ divides the interior angle $\angle BAC$ into two angles, $\angle BAD$ and $\angle DAC$ , whose measures are in the ratio $3:2$ , respectively. The measures of the interior angles $\angle B$ and $\angle C$ are in the ratio $5:4$ , respectively. If the measure of $\angle ADC$ is $104^\circ$ , what is the measure of $\angle BAC$ ?

54^\circ

72^\circ

76^\circ

90^\circ

108^\circ

Show answer & explanation

Answer: $90^\circ$

Answer

90^\circ

The correct answer is

90^\circ

. By expressing the angles in terms of variables using their ratios, we set up two independent linear equations:

3x + 5y = 104

(from the exterior angle theorem on

\triangle ABD

) and

2x + 4y = 76

(from the sum of angles in

\triangle ADC

). Solving this system yields

x = 18

. Since

\angle BAC

is composed of

\angle BAD

and

\angle DAC

, its measure is

3x + 2x = 5x = 5(18^\circ) = 90^\circ

.

Step-by-Step Solution

1

Define variables for the partitioned angles and the base angles using the given ratios.

Let the measures of

\angle BAD

and

\angle DAC

be

3x

and

2x

respectively, so that

\angle BAC = 5x

. Let the measures of

\angle B

and

\angle C

be

5y

and

4y

respectively.

Ratios express quantities as multiples of a common variable, which simplifies setting up equations.

2

Apply the exterior angle theorem to

\triangle ABD

at vertex

D

.

The exterior angle

\angle ADC = \angle BAD + \angle B \Rightarrow 104^\circ = 3x + 5y

.

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

3

Apply the angle sum theorem to

\triangle ADC

.

\angle DAC + \angle C + \angle ADC = 180^\circ \Rightarrow 2x + 4y + 104^\circ = 180^\circ \Rightarrow 2x + 4y = 76^\circ

.

The sum of the measures of the interior angles of any triangle is always

180^\circ

.

4

Solve the system of linear equations: (1)

3x + 5y = 104

and (2)

2x + 4y = 76

.

Multiply equation (1) by 2 and equation (2) by 3 to align the coefficients of

x

:

6x + 10y = 208

6x + 12y = 228

Subtract the first aligned equation from the second:

2y = 20 \Rightarrow y = 10

.
Substitute

y = 10

back into equation (2):

2x + 4(10) = 76 \Rightarrow 2x = 36 \Rightarrow x = 18

.

Solving the system of linear equations determines the values of the variables

x

and

y

.

5

Calculate the measure of

\angle BAC

.

\angle BAC = 5x = 5(18^\circ) = 90^\circ

.

We defined the total measure of

\angle BAC

as the sum of its two partitioned parts,

3x + 2x = 5x

.

Key Concept

Using triangle angle properties and exterior angle theorems to set up and solve systems of linear equations.

Question 33Question →

A rectangular garden has a straight walking path that connects two opposite corners. The length of the path is $170$ meters, and the width of the garden is $80$ meters. What is the length, in meters, of the garden?

90

120

150

188

250

Show answer & explanation

Answer: 150

Answer

The length of the garden is

150

meters.

The diagonal path, width, and length of the rectangular garden form a right triangle where the path is the hypotenuse. According to the Pythagorean theorem, the square of the length plus the square of the width equals the square of the path:

\text{length}^2 + 80^2 = 170^2

. This simplifies to

\text{length}^2 + 6,400 = 28,900

. Subtracting

6,400

from both sides gives

\text{length}^2 = 22,500

. Taking the square root of

22,500

yields

150

meters.

Step-by-Step Solution

1

Identify the right triangle formed by the length, width, and diagonal path of the garden.

The width (

80

meters) and the unknown length are the legs, while the diagonal path (

170

meters) is the hypotenuse.

The diagonal of a rectangle forms two congruent right triangles with the rectangle's sides.

2

Set up the Pythagorean equation to solve for the unknown leg.

\text{length}^2 + 80^2 = 170^2

The Pythagorean theorem states that

a^2 + b^2 = c^2

, where

c

is the hypotenuse.

3

Calculate the squares of the known lengths.

80^2 = 6,400

and

170^2 = 28,900

Evaluate the exponents to simplify the equation.

4

Isolate the squared unknown variable.

\text{length}^2 = 28,900 - 6,400 = 22,500

Subtract

6,400

from both sides of the equation.

5

Take the square root of both sides to find the length.

\text{length} = \sqrt{22,500} = 150

The square root operation reverses the squaring of the variable.

Key Concept

Applying the Pythagorean theorem to find the length of an unknown leg in a right triangle.

Estimated Time:1m 0s

Question 34Question →

In a triangle, the lengths of the sides are $x$ , $y$ , and $z$ , where $x$ , $y$ , and $z$ are integers such that $x < y < z$ . If $x = 7$ and the perimeter of the triangle is $32$ , what is the number of possible integer values for $z$ ?

Show answer & explanation

Answer: 3

Answer

There are exactly 3 possible integer values for the side length z.

The correct answer is 3. By expressing the second side as

y = 25 - z

and applying the ordering constraint

7 < 25 - z < z

, we determine that

12.5 < z < 18

. Applying the Triangle Inequality Theorem (

7 + y > z

) yields the restriction

z < 16

. Combining these conditions restricts the integer values of

z

to

\{13, 14, 15\}

, which counts to exactly 3 possible values.

Step-by-Step Solution

1

Express the side length

y

in terms of

z

.

y = 25 - z

The perimeter of the triangle is the sum of the side lengths:

x + y + z = 32

. Substituting

x = 7

gives

7 + y + z = 32

, which simplifies to

y = 25 - z

.

2

Apply the given inequality constraint

x < y < z

to find initial bounds for

z

.

12.5 < z < 18

Substituting

x = 7

and

y = 25 - z

into

x < y < z

yields

7 < 25 - z < z

. The left inequality

7 < 25 - z

simplifies to

z < 18

. The right inequality

25 - z < z

simplifies to

25 < 2z

, or

z > 12.5

.

3

Apply the Triangle Inequality Theorem to establish the final constraint on

z

.

z < 16

Since

z

is the longest side, the sum of the two shorter sides must be strictly greater than

z

:

x + y > z

. Substituting

x = 7

and

y = 25 - z

gives

7 + 25 - z > z

, which simplifies to

32 > 2z

, or

z < 16

.

4

Combine all constraints and count the valid integer values for

z

.

3 possible values (

13, 14, 15

)

Combining the bounds from the steps gives

12.5 < z < 16

. The integers satisfying this inequality are

13

,

14

, and

15

, which gives a total of 3 possible integer values.

Key Concept

Triangle Inequality Theorem and algebraic constraints on side lengths

Question 35Question →

In a right triangle, the length of the side opposite the $60^\circ$ angle is $6\sqrt{3}$ centimeters. What is the length, in centimeters, of the hypotenuse of this triangle?

Show answer & explanation

Answer: 12

Answer

The length of the hypotenuse is

12

centimeters.

In a

30^\circ-60^\circ-90^\circ

special right triangle, the sides opposite the

30^\circ

angle, the

60^\circ

angle, and the

90^\circ

(hypotenuse) angle are in the ratio

x : x\sqrt{3} : 2x

. Given that the side opposite the

60^\circ

angle is

6\sqrt{3}

centimeters, we have

x\sqrt{3} = 6\sqrt{3}

, which means

x = 6

. The hypotenuse is

2x = 2(6) = 12

centimeters.

Step-by-Step Solution

1

Determine the type of special right triangle.

A

30^\circ-60^\circ-90^\circ

right triangle.

Since the triangle is a right triangle and has a

60^\circ

angle, the remaining angle must be

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

.

2

Set up the relation for the side lengths using the ratio of a

30^\circ-60^\circ-90^\circ

triangle.

The side opposite the

60^\circ

angle is

x\sqrt{3}

centimeters, where

x

is the length of the side opposite the

30^\circ

angle.

In any

30^\circ-60^\circ-90^\circ

triangle, the side lengths are in the ratio

1 : \sqrt{3} : 2

.

3

Solve for the base variable

x

.

x = 6

We are given that the side opposite the

60^\circ

angle is

6\sqrt{3}

centimeters, so

x\sqrt{3} = 6\sqrt{3}

.

4

Calculate the length of the hypotenuse.

The hypotenuse is

2x = 2(6) = 12

centimeters.

The hypotenuse of a

30^\circ-60^\circ-90^\circ

triangle is twice the length of the shorter leg, which is

2x

.

Key Concept

Using the side length ratios of a

30^\circ-60^\circ-90^\circ

special right triangle to find missing lengths.

Question 36Question →

A triangle has two sides of length 5 and 12. The third side has a length of $x$ , where $x$ is an integer. If the perimeter of the triangle is a multiple of 5, what is the sum of all possible values of $x$ ?

13

25

21

24

39

Show answer & explanation

Answer: 21

Answer

The sum of all possible values of the third side length is 21.

To find the sum of all possible values of

x

, we first apply the Triangle Inequality Theorem. For a triangle with side lengths 5, 12, and

x

, the third side must satisfy

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

. The perimeter

P

of the triangle is given by

P = 5 + 12 + x = 17 + x

. Given that

7 < x < 17

, the perimeter must be between

17 + 7 = 24

and

17 + 17 = 34

. The only multiples of 5 within this range are 25 and 30. Setting the perimeter equal to these values gives

17 + x = 25 \implies x = 8

, and

17 + x = 30 \implies x = 13

. Both values are integers and satisfy the triangle inequality. The sum of these values is

8 + 13 = 21

.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to find the range of possible values for the third side,

x

.

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

.

The length of any side of a triangle must be strictly greater than the difference between the other two sides and strictly less than their sum.

2

Determine the expression for the perimeter of the triangle and find the bounds for the perimeter.

Perimeter

P = 5 + 12 + x = 17 + x

. Since

7 < x < 17

, the perimeter must satisfy

17 + 7 < P < 17 + 17

, which means

24 < P < 34

.

The perimeter of a triangle is the sum of its three side lengths.

3

Identify which values of the perimeter in this range are multiples of 5, and find the corresponding values of

x

.

The multiples of 5 between 24 and 34 are 25 and 30. If

P = 25

, then

17 + x = 25 \implies x = 8

. If

P = 30

, then

17 + x = 30 \implies x = 13

. Both

x = 8

and

x = 13

are integers that satisfy the initial inequality.

We must find the integer values of

x

that make the perimeter a multiple of 5.

4

Calculate the sum of all possible values of

x

.

8 + 13 = 21

.

The question asks for the sum of all valid integer values of

x

.

Key Concept

Triangle Inequality Theorem and Perimeter Calculations

Estimated Time:1m 30s

Question 37Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ and the measure of angle $A$ is $45^\circ$ . If the length of leg $AB$ is $8$ inches, what is the length, in inches, of the hypotenuse $AC$ ?

8

4\sqrt{2}

8\sqrt{2}

8\sqrt{3}

16

Show answer & explanation

Answer: $8\sqrt{2}$

Answer

The length of the hypotenuse is

8\sqrt{2}

inches.

In right triangle

ABC

, the angle measures are

45^\circ

,

45^\circ

, and

90^\circ

. The lengths of the sides of a

45^\circ

-

45^\circ

-

90^\circ

triangle are in the ratio

1 : 1 : \sqrt{2}

. Since the leg is

8

inches, the hypotenuse is

8\sqrt{2}

inches.

Step-by-Step Solution

1

Identify the type of right triangle.

Since angle

B = 90^\circ

and angle

A = 45^\circ

, angle

C

must also be

45^\circ

. This is a

45^\circ

-

45^\circ

-

90^\circ

special right triangle.

The sum of the angles in a triangle is always

180^\circ

.

2

Recall the ratio of the side lengths of a

45^\circ

-

45^\circ

-

90^\circ

triangle.

The ratio of the sides opposite the angles

45^\circ : 45^\circ : 90^\circ

is

1 : 1 : \sqrt{2}

. Thus, the hypotenuse is equal to

\text{leg} \times \sqrt{2}

.

This is a standard geometric property of isosceles right triangles.

3

Calculate the length of the hypotenuse.

Multiply the leg length of

8

inches by

\sqrt{2}

to get

8\sqrt{2}

inches.

The leg adjacent to the

45^\circ

angle is given as

8

inches.

Key Concept

Hypotenuse of a

45^\circ

-

45^\circ

-

90^\circ

special right triangle

Alternative Method

Alternatively, use the Pythagorean theorem:

AB^2 + BC^2 = AC^2

. Since it is an isosceles right triangle,

BC = AB = 8

. Thus,

8^2 + 8^2 = AC^2 \implies 64 + 64 = AC^2 \implies AC = \sqrt{128} = 8\sqrt{2}

.

Estimated Time:45s

Question 38Question →

A regular hexagon $ABCDEF$ has a side length of $8$ centimeters. A point $P$ lies on the side $CD$ such that the ratio of the length of $CP$ to the length of $PD$ is $1:3$ . What is the length, in centimeters, of the segment $AP$ ?

Show answer & explanation

Answer: 14

Answer

The length of the segment

AP

is

14

centimeters.

The correct answer is

14

. Dropping a perpendicular from

P

to the main diagonal

AD

creates a

30^\circ-60^\circ-90^\circ

triangle

\triangle PHD

with hypotenuse

PD = 6

. The legs are

DH = 3

and

PH = 3\sqrt{3}

. This leaves

AH = 13

. Applying the Pythagorean Theorem to the right triangle

\triangle AHP

with legs

13

and

3\sqrt{3}

yields

AP = \sqrt{13^2 + (3\sqrt{3})^2} = 14

.

Step-by-Step Solution

1

Determine the length of the main diagonal

AD

of the regular hexagon.

AD = 16

cm

In a regular hexagon with side length

s

, the main diagonal connecting opposite vertices has a length of

2s

. Given

s = 8

, we find

AD = 2 \times 8 = 16

.

2

Calculate the length of the segment

PD

on the side

CD

.

PD = 6

cm

The point

P

divides the side

CD

of length

8

in the ratio

CP:PD = 1:3

. Thus,

PD = \frac{3}{1+3} \times 8 = 6

.

3

Identify the angles and type of triangle formed by dropping a perpendicular from

P

to diagonal

AD

.

\triangle PHD

is a

30^\circ-60^\circ-90^\circ

right triangle.

The diagonal

AD

bisects the interior angle

\angle CDE = 120^\circ

of the regular hexagon, making

\angle ADC = 60^\circ

. Since

PH \perp AD

, the triangle

\triangle PHD

has angles

90^\circ

,

60^\circ

, and

30^\circ

.

4

Find the lengths of the legs

DH

and

PH

of the special right triangle

\triangle PHD

.

DH = 3

cm and

PH = 3\sqrt{3}

cm

Using the ratios of a

30^\circ-60^\circ-90^\circ

triangle with hypotenuse

PD = 6

, the leg adjacent to the

60^\circ

angle is

DH = 6 \cos(60^\circ) = 3

, and the leg opposite to the

60^\circ

angle is

PH = 6 \sin(60^\circ) = 3\sqrt{3}

.

5

Calculate the length of the segment

AH

.

AH = 13

cm

Since

H

lies on the diagonal

AD

, we subtract the length of

DH

from the total length of the diagonal:

AH = AD - DH = 16 - 3 = 13

.

6

Apply the Pythagorean Theorem to the right triangle

\triangle AHP

to find the length of

AP

.

AP = 14

cm

In the right triangle

\triangle AHP

with legs

AH = 13

and

PH = 3\sqrt{3}

, the hypotenuse is

AP = \sqrt{AH^2 + PH^2} = \sqrt{13^2 + (3\sqrt{3})^2} = \sqrt{169 + 27} = \sqrt{196} = 14

.

Key Concept

Applying special right triangle ratios and the Pythagorean Theorem in multi-step geometric figures.

Question 39Question →

An isosceles triangle has two sides of length $5$ and $11$ . A second triangle has side lengths of $12$ , $18$ , and $d$ , where $d$ is an integer. If $d$ is equal to the perimeter of the first triangle, what is the perimeter of the second triangle?

41

46

51

57

63

Show answer & explanation

Answer: 57

Answer

57

To find the perimeter of the second triangle, we must first determine the value of

d

, which is the perimeter of the first triangle. The first triangle is isosceles with two sides of length

5

and

11

. By the Triangle Inequality Theorem, the sum of any two side lengths must exceed the third. A triangle with sides

5, 5, 11

is impossible because

5 + 5 = 10 < 11

. Thus, the sides of the first triangle must be

11, 11, 5

, giving a perimeter of

11 + 11 + 5 = 27

. This means

d = 27

. The second triangle has sides of length

12

,

18

, and

27

. Since

12 + 18 = 30 > 27

, this is a valid triangle. Its perimeter is

12 + 18 + 27 = 57

.

Step-by-Step Solution

1

Determine the possible side lengths of the first isosceles triangle.

The sides must be

11

,

11

, and

5

.

An isosceles triangle has two equal sides. The side lengths must be either

5, 5, 11

or

11, 11, 5

. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. If the sides were

5, 5, 11

, then

5 + 5 = 10 < 11

, which violates this theorem. Thus, the only valid side lengths are

11

,

11

, and

5

(since

5 + 11 = 16 > 11

).

2

Calculate the perimeter of the first triangle to find the value of

d

.

d = 27

The perimeter of the first triangle is the sum of its three sides:

11 + 11 + 5 = 27

. Since

d

is equal to this perimeter,

d = 27

.

3

Verify that a triangle with side lengths

12

,

18

, and

27

is valid.

The triangle is valid.

We check the Triangle Inequality Theorem:

12 + 18 = 30 > 27

,

12 + 27 = 39 > 18

, and

18 + 27 = 45 > 12

. Since all inequalities hold, the second triangle is valid.

4

Calculate the perimeter of the second triangle.

Perimeter =

57

The perimeter of the second triangle is the sum of its side lengths:

12 + 18 + 27 = 57

.

Key Concept

Triangle Inequality Theorem and Isosceles Triangle Properties

Estimated Time:2m 0s

Question 40Question →

In right triangle $ABC$ , the hypotenuse $AC$ has a length of $15$ centimeters, and leg $AB$ has a length of $9$ centimeters. What is the length, in centimeters, of leg $BC$ ?

Show answer & explanation

Answer: 12

Answer

The length of leg

BC

is

12

centimeters.

The length of leg

BC

is found using the Pythagorean Theorem,

AB^2 + BC^2 = AC^2

. Substituting the given values gives

9^2 + BC^2 = 15^2

, which simplifies to

81 + BC^2 = 225

. Subtracting 81 from both sides yields

BC^2 = 144

. Taking the square root of 144 gives the correct length of 12 centimeters.

Step-by-Step Solution

1

Identify the given dimensions and apply the Pythagorean Theorem.

AB^2 + BC^2 = AC^2

For any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

2

Substitute the known values

AB = 9

and

AC = 15

into the equation.

9^2 + BC^2 = 15^2

The hypotenuse

AC

is the side opposite the right angle, and

AB

is one of the legs.

3

Simplify the squared terms.

81 + BC^2 = 225

Squaring 9 yields 81, and squaring 15 yields 225.

4

Isolate the unknown term by subtracting 81 from both sides.

BC^2 = 144

Subtracting 81 from both sides isolates

BC^2

on the left side of the equation.

5

Take the square root of both sides to solve for the leg length.

BC = 12

Taking the square root of 144 gives the side length, which must be positive.

Key Concept

Pythagorean Theorem

Plane Geometry

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution