Triangles and Pythagorean Theorem Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

Question 21Question →

An isosceles triangle has two equal sides that each measure $13$ inches and a base that measures $10$ inches. What is the length of the altitude (height) drawn to the base?

8

inches

36

inches

12

inches

18

inches

Show answer & explanation

Answer: $12$ inches

Answer

The length of the altitude is

12

inches.

Drawing an altitude to the base of an isosceles triangle splits it into two identical right triangles. The base of

10

inches is bisected into two

5

-inch segments. For one of these right triangles, the base leg is

5

inches and the hypotenuse (the equal side) is

13

inches. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we set up the equation

h^2 + 5^2 = 13^2

. Solving for

h^2

gives

169 - 25 = 144

. Taking the square root of

144

yields an altitude of

12

inches.

Step-by-Step Solution

1

Identify the properties of an altitude in an isosceles triangle.

The altitude bisects the

10

-inch base, creating two equal segments of

5

inches each.

This splits the isosceles triangle into two congruent right triangles, allowing the use of the Pythagorean theorem.

2

Set up the Pythagorean theorem (

a^2 + b^2 = c^2

) for one of the right triangles.

h^2 + 5^2 = 13^2

, where

h

is the altitude,

5

is the base leg, and

13

is the hypotenuse.

The equal side of the isosceles triangle serves as the hypotenuse of the newly formed right triangle.

3

Square the known values.

h^2 + 25 = 169

.

To begin isolating the unknown altitude squared (

h^2

).

4

Solve for the altitude.

h^2 = 169 - 25 = 144

, so

h = \sqrt{144} = 12

.

Taking the square root of

144

gives the final length of the altitude.

Key Concept

Using the Pythagorean theorem to find the altitude of an isosceles triangle

Question 22Question →

A right triangle has a hypotenuse that measures $20$ centimeters and a base that measures $12$ centimeters. What is the area of this triangle?

48\text{ cm}^2

120\text{ cm}^2

96\text{ cm}^2

192\text{ cm}^2

Show answer & explanation

Answer: $96\text{ cm}^2$

Answer

The area of the right triangle is

96\text{ cm}^2

.

To find the area of a right triangle, you need both the base and the height (the two legs). The Pythagorean theorem (

a^2 + b^2 = c^2

) reveals the missing height is

16

cm because

16^2 + 12^2 = 20^2

. Using the area formula (

A = \frac{1}{2}bh

), we multiply

\frac{1}{2} \times 12 \times 16

to get

96\text{ cm}^2

.

Step-by-Step Solution

1

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing height (the other leg).

a^2 + 12^2 = 20^2

The area formula requires the base and the height. We are given the hypotenuse (

20

) and the base (

12

).

2

Solve for the missing height (

a

).

a^2 + 144 = 400 \Rightarrow a^2 = 256 \Rightarrow a = 16

centimeters.

Subtracting

144

from

400

gives

256

, and the square root of

256

is

16

.

3

Calculate the area of the triangle using the formula

A = \frac{1}{2}bh

.

A = \frac{1}{2} \times 12 \times 16 = 96\text{ cm}^2

.

Substituting the base (

12

) and the newly found height (

16

) into the area formula gives the final answer.

Key Concept

Applying the Pythagorean theorem to find a missing side length, then calculating the triangle's area.

Estimated Time:1m 0s

Question 23Question →

A rectangular swimming pool has a diagonal length of $20\text{ meters}$ and a width of $12\text{ meters}$ . What is the perimeter of the swimming pool?

16\text{ meters}

48\text{ meters}

56\text{ meters}

64\text{ meters}

Show answer & explanation

Answer: $56\text{ meters}$

Answer

56\text{ meters}

The diagonal of a rectangle creates a right triangle where the diagonal is the hypotenuse. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the missing length is

\sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16\text{ meters}

. The perimeter of the rectangle is the total distance around the outside, which is calculated as

2 \times (\text{width} + \text{length}) = 2 \times (12 + 16) = 2 \times 28 = 56\text{ meters}

.

Step-by-Step Solution

1

Identify the geometric relationship between the dimensions.

The width, length, and diagonal of a rectangle form a right triangle where the diagonal is the hypotenuse.

Rectangles have right angles at all corners, allowing the use of the Pythagorean theorem.

2

Use the Pythagorean theorem to find the length of the pool.

12^2 + \text{length}^2 = 20^2

, which becomes

144 + \text{length}^2 = 400

. Subtracting

144

gives

\text{length}^2 = 256

, so the length is

16\text{ meters}

.

The Pythagorean theorem (

a^2 + b^2 = c^2

) allows us to find a missing side of a right triangle when two sides are known.

3

Calculate the perimeter of the rectangle.

Perimeter =

2 \times (\text{width} + \text{length}) = 2 \times (12 + 16) = 2 \times 28 = 56\text{ meters}

.

The question asks for the perimeter of the entire rectangular pool, which requires combining all four outer sides.

Key Concept

Applying the Pythagorean theorem to find missing dimensions before calculating perimeter.

Estimated Time:1m 0s

Question 24Question →

Provide the missing numerical values to complete the calculations for the stage construction.

Fill in the blanks below

A theater crew is reinforcing a rectangular stage that measures

16

feet in width and

30

feet in length. They place a diagonal steel support beam underneath, running directly from one corner to the opposite corner. By applying the Pythagorean theorem, the length of this diagonal support beam is calculated to be feet. Consequently, the total perimeter of the right triangle formed by the width, length, and the new diagonal beam is feet.

Show answer & explanation

Answer

The diagonal support beam is 34 feet long, and the total perimeter of the right triangle is 80 feet.

The width and length of the rectangular stage act as the two legs of a right triangle. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known legs:

16^2 + 30^2 = 256 + 900 = 1156

. The square root of

1156

is

34

, so the diagonal hypotenuse is

34

feet. The perimeter of this triangle is the sum of its three sides:

16 + 30 + 34 = 80

feet.

Step-by-Step Solution

1

Identify the lengths of the two legs of the right triangle formed by the stage's dimensions.

Leg a = 16 feet, Leg b = 30 feet.

The width and length of a rectangle meet at a 90-degree angle, serving as the legs of the right triangle.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to calculate the hypotenuse.

16^2 + 30^2 = 256 + 900 = 1156

. Taking the square root,

c = \sqrt{1156} = 34

feet.

The diagonal beam forms the hypotenuse of the right triangle spanning opposite corners.

3

Calculate the perimeter of the newly formed right triangle.

Perimeter =

16 + 30 + 34 = 80

feet.

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

Key Concept

Applying the Pythagorean theorem to find a missing hypotenuse, and calculating the perimeter of a right triangle.

Question 25Question →

A custom corner desk is shaped like a right triangle so it fits perfectly against two perpendicular walls. The sections of the desk that touch the walls measure $24\text{ inches}$ and $10\text{ inches}$ in length. What is the length of the front edge of the desk in inches?

Show answer & explanation

Answer: 26

Answer

26

Because the walls meet at a perpendicular right angle, the corner desk forms a right triangle. The edges touching the walls act as the legs (

a = 24

,

b = 10

), and the front edge is the hypotenuse (

c

). Using the Pythagorean Theorem (

a^2 + b^2 = c^2

), we square both legs to get

576

and

100

. Adding these together gives

676

. The square root of

676

is

26

, which is the exact length of the front edge in inches.

Step-by-Step Solution

1

Identify the known sides of the right triangle

Legs are

a = 24

and

b = 10

.

The walls are perpendicular, forming a

90^{\circ}

angle. Therefore, the edges touching the walls are the legs of a right triangle.

2

Set up the Pythagorean Theorem

24^2 + 10^2 = c^2

The Pythagorean Theorem (

a^2 + b^2 = c^2

) is the standard formula used to find the missing hypotenuse (

c

) when both legs are known.

3

Square the lengths of the legs

576 + 100 = c^2

Evaluating the exponents is the required next step in the order of operations.

4

Add the squared values

676 = c^2

Combining the squares of the legs gives the square of the hypotenuse.

5

Find the square root of the sum

c = 26

Taking the square root of

676

isolates

c

, revealing the true length of the desk's front edge.

Key Concept

Using the Pythagorean Theorem (

a^2 + b^2 = c^2

) to calculate the length of a hypotenuse.

Alternative Method

You can solve this rapidly without large calculations by recognizing special Pythagorean triples. This triangle is a scaled-up version of the fundamental

5-12-13

right triangle. Multiplying each side of a

5-12-13

triangle by

2

gives side lengths of

10

,

24

, and

26

. Since the legs are

10

and

24

, the hypotenuse must be

26

.

Estimated Time:45s

Question 26Question →

A right-angled triangular bracket is used to support a shelf. The longest side (hypotenuse) of the bracket has a length represented by the expression $2x + 1$ inches, and the horizontal leg has a length represented by $3x - 6$ inches. If $x = 6$ , what is the length of the vertical leg?

1

5

25

313

Show answer & explanation

Answer: 5

Answer

The vertical leg has a length of

5

inches.

After substituting

x = 6

, the hypotenuse is

13

inches and the horizontal leg is

12

inches. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we get

b^2 + 12^2 = 13^2

. Solving for

b

yields

b^2 = 25

, which means

b = 5

.

Step-by-Step Solution

1

Substitute

x = 6

into the expression for the hypotenuse.

2(6) + 1 = 12 + 1 = 13\text{ inches}

To find the numerical length of the longest side.

2

Substitute

x = 6

into the expression for the horizontal leg.

3(6) - 6 = 18 - 6 = 12\text{ inches}

To find the numerical length of the given leg.

3

Apply the Pythagorean theorem to find the vertical leg (let's call it

b

).

b^2 + 12^2 = 13^2

The Pythagorean theorem relates the three sides of a right triangle:

a^2 + b^2 = c^2

.

4

Solve for

b

.

b^2 + 144 = 169 \Rightarrow b^2 = 169 - 144 = 25 \Rightarrow b = 5\text{ inches}

To calculate the final missing side length.

Key Concept

To find a missing leg in a right triangle when given the hypotenuse and the other leg, substitute the known values into the Pythagorean theorem (

a^2 + b^2 = c^2

) and solve for the unknown leg by subtracting the squares and taking the square root.

Alternative Method

Recognize that

5

,

12

, and

13

form a common Pythagorean triple. Once the hypotenuse (

13

) and one leg (

12

) are calculated, the other leg must immediately be

5

.

Estimated Time:1m 15s

Question 27Question →

Read the scenario and provide the missing numerical value to complete the sentence.

Fill in the blanks below

A guy wire is securely attached to a vertical radio mast at a height of

60\text{ feet}

above the level ground. The other end of the wire is anchored to the ground exactly

25\text{ feet}

from the base of the mast. Assuming the mast is perfectly perpendicular to the ground, the length of the guy wire is feet.

Show answer & explanation

Answer

65

The mast, the ground, and the guy wire form a right triangle where the mast and the ground act as the two legs, and the guy wire is the hypotenuse. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the given leg lengths:

25^2 + 60^2 = c^2

. This simplifies to

625 + 3600 = 4225

. The square root of

4225

is

65

, meaning the guy wire is

65\text{ feet}

long. Additionally, recognizing that the side lengths (

25, 60, 65

) are a multiple of the common

5-12-13

Pythagorean triple (each multiplied by

5

) provides a quick way to solve the problem without large calculations.

Step-by-Step Solution

1

Identify the shape formed by the scenario.

A right triangle is formed by the vertical mast, the level ground, and the diagonal guy wire.

The mast is perpendicular to the ground, creating a 90-degree angle.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, where

a = 25

and

b = 60

.

The legs of the right triangle are the distance along the ground (25 ft) and the height up the mast (60 ft). We need to find the hypotenuse,

c

.

3

Calculate the squares of the leg lengths.

25^2 = 625

and

60^2 = 3600

.

Squaring the values is required before adding them to find the square of the hypotenuse.

4

Add the squared values together.

625 + 3600 = 4225

.

This sum represents

c^2

, the square of the guy wire's length.

5

Find the square root of the sum to determine the final length.

\sqrt{4225} = 65

.

Taking the square root of

c^2

gives the actual length of the hypotenuse,

c

.

Key Concept

Pythagorean Theorem Application

Question 28Question →

In a right triangle, the length of the hypotenuse is denoted by $c$ , and the lengths of the two legs are denoted by $a$ and $b$ . If $a = 8\text{ cm}$ and $c = 17\text{ cm}$ , what is the length of leg $b$ ?

9\text{ cm}

225\text{ cm}

15\text{ cm}

\sqrt{353}\text{ cm}

Show answer & explanation

Answer: $15\text{ cm}$

Answer

The length of leg

b

is

15\text{ cm}

.

The Pythagorean theorem states that for any right triangle, the sum of the squares of the legs equals the square of the hypotenuse (

a^2 + b^2 = c^2

). By substituting

a = 8

and

c = 17

, the equation becomes

64 + b^2 = 289

. Subtracting

64

from both sides gives

b^2 = 225

. Taking the square root of

225

yields a leg length of

15\text{ cm}

.

Step-by-Step Solution

1

Identify the known and unknown values in the Pythagorean theorem formula.

Formula:

a^2 + b^2 = c^2

. Known:

a = 8

and

c = 17

. Unknown:

b

.

To ensure values are placed in the correct positions, specifically identifying

17

as the hypotenuse (

c

).

2

Substitute the known values into the equation and square them.

8^2 + b^2 = 17^2

becomes

64 + b^2 = 289

.

To simplify the equation before isolating the unknown variable.

3

Isolate

b^2

by subtracting

64

from both sides of the equation.

b^2 = 289 - 64

, which simplifies to

b^2 = 225

.

To get the squared unknown variable by itself on one side of the equals sign.

4

Take the square root of both sides to solve for

b

.

b = \sqrt{225} = 15\text{ cm}

.

To remove the exponent and find the final linear length of the leg.

Key Concept

Pythagorean Theorem and Variable Substitution

Estimated Time:1m 0s

Question 29Question →

A painter leans a $25\text{-foot}$ ladder against a vertical wall. The base of the ladder is placed exactly $7\text{ feet}$ away from the base of the wall on level ground. How high up the wall, in feet, does the top of the ladder reach?

Show answer & explanation

Answer: 24

Answer

The top of the ladder reaches 24 feet up the wall.

The correct answer is derived by modeling the scenario as a right triangle. The ladder acts as the hypotenuse (

c = 25

) and the distance along the ground acts as one leg (

a = 7

). Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we solve for the missing height (

b

). Substituting the values gives

7^2 + b^2 = 25^2

, which simplifies to

49 + b^2 = 625

. Subtracting 49 yields

b^2 = 576

. The square root of 576 is 24, meaning the ladder reaches 24 feet up the wall.

Step-by-Step Solution

1

Set up the right triangle relationship.

a^2 + b^2 = c^2

, where

c = 25

and

a = 7

The wall is perpendicular to the ground, forming a right triangle with the ladder acting as the hypotenuse.

2

Substitute known values into the Pythagorean theorem.

7^2 + b^2 = 25^2

We know the hypotenuse (

25

) and one leg (

7

), and we need to find the other leg (

b

).

3

Square the known numbers.

49 + b^2 = 625

Evaluating the exponents is required before isolating the variable term.

4

Isolate the variable term.

b^2 = 625 - 49 = 576

Subtracting

49

from both sides leaves

b^2

by itself.

5

Solve for the height.

b = \sqrt{576} = 24

Taking the square root of both sides gives the length of the missing side.

Key Concept

Using the Pythagorean theorem to find a missing leg of a right triangle in a real-world scenario.

Question 30Question →

A drone takes off from a launch pad and flies $6\text{ kilometers}$ due south. It then turns and flies due west. If the drone's straight-line distance from the launch pad is now exactly $10\text{ kilometers}$ , how many kilometers did it fly due west?

4

8

16

64

Show answer & explanation

Answer: 8

Answer

The drone flew 8 kilometers due west.

The drone's path forms a right triangle with a known leg of

6\text{ km}

, an unknown leg

x

, and a hypotenuse of

10\text{ km}

. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known values:

6^2 + x^2 = 10^2

. This simplifies to

36 + x^2 = 100

. Subtracting

36

from both sides gives

x^2 = 64

. Taking the square root of both sides yields

x = 8

. Therefore, the drone flew

8\text{ kilometers}

due west.

Step-by-Step Solution

1

Identify the geometric shape formed by the drone's flight path.

The flight path forms a right-angled triangle where the southward and westward paths are the legs, and the straight-line distance is the hypotenuse.

The cardinal directions South and West are perpendicular to each other, forming a 90-degree angle.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

6^2 + b^2 = 10^2

The southward distance is one leg (

a = 6

), the westward distance is the unknown leg (

b

), and the straight-line distance is the hypotenuse (

c = 10

).

3

Solve the equation for the unknown leg.

36 + b^2 = 100 \rightarrow b^2 = 64 \rightarrow b = 8

Squaring the known sides and subtracting gives the square of the unknown leg. Taking the square root gives the final distance.

Key Concept

Using the Pythagorean theorem to find a missing leg of a right triangle in a real-world navigation scenario.

Estimated Time:1m 0s

Question 31Question →

Determine the missing measurement to complete the statement below.

Fill in the blanks below

A museum exhibit features a custom glass panel shaped exactly like a right triangle. The longest edge of the glass panel measures

29\text{ inches}

, and the shortest edge measures

20\text{ inches}

. The length of the third edge is inches.

Show answer & explanation

Answer

21

The correct answer is

21

. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), the square of the hypotenuse (

29^2 = 841

) minus the square of the known leg (

20^2 = 400

) equals the square of the unknown leg (

441

). The square root of

441

is

21

.

Step-by-Step Solution

1

Identify the given parts of the right triangle.

Hypotenuse

c = 29

, and leg

a = 20

.

The longest edge of a right triangle is always the hypotenuse.

2

Set up the Pythagorean theorem equation.

20^2 + b^2 = 29^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) connects the lengths of the legs to the hypotenuse.

3

Square the known side lengths.

400 + b^2 = 841

20 \times 20 = 400

and

29 \times 29 = 841

.

4

Isolate the unknown variable

b^2

.

b^2 = 841 - 400 = 441

Subtract

400

from both sides to find the square of the missing leg.

5

Solve for

b

by taking the square root.

b = \sqrt{441} = 21

The square root of

441

gives the length of the third edge in inches.

Key Concept

The Pythagorean Theorem (

a^2 + b^2 = c^2

) allows you to find an unknown leg of a right triangle by subtracting the square of the known leg from the square of the hypotenuse.

Question 32Question →

A rectangular television screen has a horizontal width of $20\text{ inches}$ and a vertical height of $15\text{ inches}$ . What is the length of the diagonal of the screen?

5\text{ inches}

\sqrt{175}\text{ inches}

25\text{ inches}

35\text{ inches}

Show answer & explanation

Answer: $25\text{ inches}$

Answer

The correct length of the diagonal is

25\text{ inches}

.

The diagonal of the screen forms the hypotenuse of a right triangle with the width and height acting as the two legs. By applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we calculate

20^2 + 15^2 = 400 + 225 = 625

. Taking the square root of 625 gives a diagonal length of 25 inches.

Step-by-Step Solution

1

Identify the shape and the relationship between the given measurements.

The width (

20\text{ inches}

) and height (

15\text{ inches}

) form the legs of a right triangle, and the diagonal represents the hypotenuse.

A rectangle's corners are right angles, so drawing a diagonal line creates two identical right triangles.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing hypotenuse.

15^2 + 20^2 = c^2

The Pythagorean theorem relates the lengths of the legs (

a

and

b

) to the hypotenuse (

c

) in any right triangle.

3

Square the lengths of the legs and add them together.

225 + 400 = 625

Following the order of operations, exponents must be evaluated before addition.

4

Take the square root of the sum to find the length of the diagonal.

\sqrt{625} = 25\text{ inches}

To solve for

c

from

c^2

, we must find the principal square root.

Key Concept

Using the Pythagorean theorem to find the hypotenuse (diagonal) of a right triangle formed within a rectangle.

Estimated Time:1m 0s

Question 33Question →

The entrance of a camping tent is shaped like an isosceles triangle. The base of the tent resting on the ground is $16\text{ feet}$ wide, and the two slanted sides are each $17\text{ feet}$ long. What is the maximum height of the tent, in feet, from the ground to the peak?

Show answer & explanation

Answer: 15

Answer

15

By dividing the isosceles triangle in half, we form a right triangle with a hypotenuse of 17 and a base leg of 8. Applying the Pythagorean theorem (

8^2 + h^2 = 17^2

) gives

64 + h^2 = 289

, resulting in

h^2 = 225

. The square root of 225 is 15, which is the maximum height of the tent.

Step-by-Step Solution

1

Divide the base of the isosceles triangle in half to find the base of the inner right triangle.

16 \div 2 = 8\text{ feet}

The height (altitude) of an isosceles triangle bisects the base, creating a right angle necessary for the Pythagorean theorem.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

h^2 + 8^2 = 17^2

The vertical height (

h

) and half of the base (

8

) are the legs of the right triangle, while the slanted side (

17

) is the hypotenuse.

3

Square the known values.

h^2 + 64 = 289

Simplifying the terms makes it possible to solve for the unknown.

4

Subtract 64 from both sides to isolate

h^2

.

h^2 = 225

Standard algebraic process for solving a one-variable equation.

5

Take the square root of both sides to find

h

.

h = 15

The inverse operation of squaring gives the final linear dimension of the height.

Key Concept

Using the properties of isosceles triangles alongside the Pythagorean theorem to find an unknown altitude

Question 34Question →

A right-angled triangular warning sign is being designed for a factory floor. The longest side of the sign measures $17\text{ inches}$ , and one of its perpendicular sides measures $8\text{ inches}$ . What is the total area of the front face of the sign?

36\text{ square inches}

40\text{ square inches}

60\text{ square inches}

68\text{ square inches}

Show answer & explanation

Answer: $60\text{ square inches}$

Answer

60\text{ square inches}

To find the area of a right triangle, the lengths of both perpendicular legs must be known. Using the Pythagorean theorem, the missing leg is calculated as

15\text{ inches}

(since

\sqrt{17^2 - 8^2} = 15

). The area is then found by multiplying one half by the product of the two legs (

0.5 \times 8 \times 15 = 60

).

Step-by-Step Solution

1

Identify the given sides of the right triangle.

The hypotenuse (longest side,

c

) is

17\text{ inches}

and one leg (

a

) is

8\text{ inches}

.

The area formula for a right triangle requires the lengths of both perpendicular legs, not the hypotenuse.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing leg (

b

).

8^2 + b^2 = 17^2

, which simplifies to

64 + b^2 = 289

. Solving for

b^2

gives

225

, so

b = 15\text{ inches}

.

Both perpendicular legs are needed to establish the base and height of the triangle.

3

Calculate the area of the triangle using the formula Area =

0.5 \times \text{base} \times \text{height}

.

Area =

0.5 \times 8 \times 15 = 60\text{ square inches}

.

The two legs of a right triangle serve as its base and height.

Key Concept

Pythagorean Theorem and Triangle Area

Question 35Question →

Fill in the blanks with the correct numerical values based on the geometric properties described.

Fill in the blanks below

A city park features a right-angled triangular garden. The longest side of the garden measures

34\text{ yards}

, and one of the shorter sides is

16\text{ yards}

. The length of the other shorter side is yards, and the area of the garden is square yards.

Show answer & explanation

Answer

The missing shorter side is 30 yards, and the area of the garden is 240 square yards.

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) with a hypotenuse of

34

and a leg of

16

, the missing leg is

\sqrt{34^2 - 16^2} = \sqrt{1156 - 256} = \sqrt{900} = 30\text{ yards}

. The area is found by multiplying half of the base by the height:

\frac{1}{2} \times 16 \times 30 = 240\text{ square yards}

.

Step-by-Step Solution

1

Identify the given sides in relation to a right-angled triangle.

The hypotenuse (longest side)

c = 34\text{ yards}

, and one leg

a = 16\text{ yards}

. The other leg

b

is unknown.

Correctly identifying the hypotenuse is necessary to set up the Pythagorean theorem.

2

Use the Pythagorean theorem to find the length of the unknown leg.

a^2 + b^2 = c^2 \Rightarrow 16^2 + b^2 = 34^2 \Rightarrow 256 + b^2 = 1156 \Rightarrow b^2 = 900 \Rightarrow b = 30

.

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

3

Calculate the area of the triangular garden.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16 \times 30 = 240

.

The area of a right triangle is half the product of the lengths of its two perpendicular legs.

Key Concept

Using the Pythagorean theorem to find a missing leg, and applying the triangle area formula.

Question 36Question →

A naval patrol vessel leaves port and sails due South for $10\text{ nautical miles}$ . It then changes course and sails due West. After completing this leg of the journey, the vessel's radar indicates it is exactly $26\text{ nautical miles}$ in a straight line from the port. How many nautical miles did the vessel travel due West?

16

24

36

576

Show answer & explanation

Answer: $24$

Answer

The vessel traveled

24

nautical miles due West.

The path of the vessel forms a right triangle. The straight-line distance from the port is the hypotenuse (

26\text{ nautical miles}

), and the southward distance is one leg (

10\text{ nautical miles}

). Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known values:

10^2 + b^2 = 26^2

. This simplifies to

100 + b^2 = 676

. Subtracting

100

from both sides gives

b^2 = 576

. Taking the square root of

576

yields

24\text{ nautical miles}

.

Step-by-Step Solution

1

Identify that the vessel's path forms a right triangle.

The southward path (

10

) and westward path (

x

) are the legs, and the straight-line distance (

26

) is the hypotenuse.

South and West directions are perpendicular, creating a 90-degree angle.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

).

10^2 + x^2 = 26^2

This formula relates the lengths of the legs to the hypotenuse in a right triangle.

3

Square the known values.

100 + x^2 = 676

To begin isolating the unknown variable

x

.

4

Subtract

100

from both sides of the equation.

x^2 = 576

To completely isolate the

x^2

term.

5

Take the square root of both sides.

x = 24

To find the final distance traveled due West.

Key Concept

Applying the Pythagorean theorem to find the missing leg of a right triangle.

Estimated Time:1m 0s

Question 37Question →

The floor of a rectangular storage room has a diagonal measurement of $39$ feet and a width of $15$ feet. What is the total area of the floor in square feet?

Show answer & explanation

Answer: 540

Answer

540

The storage room floor forms a right triangle with the 39-foot diagonal as the hypotenuse and the 15-foot width as a leg. The missing length is calculated to be 36 feet using the Pythagorean theorem. Multiplying this 36-foot length by the 15-foot width yields a total area of 540 square feet.

Step-by-Step Solution

1

Identify the right triangle formed by the floor dimensions

A right triangle is formed with the diagonal (39 feet) as the hypotenuse and the width (15 feet) as one leg.

The diagonal of any rectangle splits it into two identical right triangles.

2

Find the missing length of the floor

The length is 36 feet.

Using the Pythagorean theorem (

a^2 + b^2 = c^2

),

15^2 + b^2 = 39^2

. This simplifies to

225 + b^2 = 1521

, so

b^2 = 1296

and

b = 36

. Alternatively, recognize the 5-12-13 Pythagorean triple multiplied by 3.

3

Calculate the area of the rectangle

Area = 540 square feet.

The area of a rectangle is its length multiplied by its width (

36 \times 15 = 540

).

Key Concept

Using the Pythagorean theorem to find unknown dimensions for area calculations.

Alternative Method

Instead of calculating

39^2

and

15^2

manually, recognize that the given sides are multiples of the common 5-12-13 Pythagorean triple (

3 \times 5 = 15

and

3 \times 13 = 39

). This immediately gives the missing length as

3 \times 12 = 36

.

Estimated Time:1m 0s

Question 38Question →

A canvas sail is shaped like a right triangle. The longest side of the sail measures $15\text{ feet}$ , and the bottom edge measures $9\text{ feet}$ . What is the total area of the canvas sail?

54\text{ square feet}

67.5\text{ square feet}

108\text{ square feet}

27\text{ square feet}

Show answer & explanation

Answer: $54\text{ square feet}$

Answer

54\text{ square feet}

The area of a right triangle is half the product of its legs. First, the Pythagorean theorem (

a^2 + b^2 = c^2

) must be used to find the missing perpendicular height (the second leg). The longest side is the hypotenuse (

15

), and the bottom edge is a leg (

9

). Solving for the missing height gives

\sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12\text{ feet}

. The area is then found by using the formula

\frac{1}{2} \times \text{base} \times \text{height}

, which equals

\frac{1}{2} \times 9 \times 12 = 54\text{ square feet}

.

Step-by-Step Solution

1

Identify the known sides of the right triangle.

The hypotenuse (longest side) is

15\text{ feet}

and the base (one leg) is

9\text{ feet}

.

The Pythagorean theorem requires knowing two sides of a right triangle to find the third.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing height (the other leg).

Substituting the known values gives

9^2 + b^2 = 15^2

, which simplifies to

81 + b^2 = 225

. Solving for

b^2

gives

144

, meaning the height is

\sqrt{144} = 12\text{ feet}

.

The height of the right triangle is necessary to calculate its total area.

3

Calculate the area of the right triangle.

Area =

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 = 54\text{ square feet}

.

The standard triangle area formula provides the final answer.

Key Concept

Using the Pythagorean theorem to find a missing side length before calculating the area of a right triangle.

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