# GED Math : Pythagorean Theorem

## Example Questions

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### Example Question #1 : Pythagorean Theorem

The two legs of a right triangle measure 30 and 40. What is its perimeter?

Explanation:

By the Pythagorean Theorem, if  are the legs of a right triangle and  is its hypotenuse,

Substitute  and solve for :

The perimeter of the triangle is

### Example Question #2 : Pythagorean Theorem

A right triangle has legs 30 and 40. Give its perimeter.

Explanation:

The hypotenuse of the right triangle can be calculated using the Pythagorean theorem:

Add the three sides:

### Example Question #3 : Pythagorean Theorem

A right triangle has one leg measuring 14 inches; its hypotenuse is 50 inches. Give its perimeter.

Explanation:

The Pythagorean Theorem can be used to derive the length of the second leg:

inches

Add the three sides to get the perimeter.

inches.

### Example Question #4 : Pythagorean Theorem

Whicih of the following could be the lengths of the sides of a right triangle?

10 inches, 1 foot, 14 inches

15 inches, 3 feet, 40 inches

2 feet, 32 inches, 40 inches

7 inches, 2 feet, 30 inches

2 feet, 32 inches, 40 inches

Explanation:

A triangle is right if and only if it satisfies the Pythagorean relationship

where  is the measure of the longest side and  are the other two sidelengths. We test each of the four sets of lengths, remembering to convert feet to inches by multiplying by 12.

7 inches, 2 feet, 30 inches:

2 feet is equal to 24 inches. The relationship to be tested is

- False

10 inches, 1 foot, 14 inches:

1 foot is equal to 12 inches. The relationship to be tested is

- False

15 inches, 3 feet, 40 inches:

3 feet is equal to 36 inches. The relationship to be tested is

- False

2 feet, 32 inches, 40 inches:

2 feet is equal to 24 inches. The relationship to be tested is

- True

The correct choice is 2 feet, 32 inches, 40 inches.

### Example Question #5 : Pythagorean Theorem

An isosceles right triangle has hypotenuse 80 inches. Give its perimeter. (If not exact, round to the nearest tenth of an inch.)

Explanation:

Each leg of an isosceles right triangle has length that is the length of the hypotenuse divided by . The hypotenuse has length 80, so each leg has length

.

The perimeter is the sum of the three sides:

inches.To the nearest tenth:

inches.

### Example Question #6 : Pythagorean Theorem

Note: Figure NOT drawn to scale.

Refer to the above diagram. . Give the perimeter of Quadrilateral

Explanation:

The perimeter of Quadrilateral  is the sum of the lengths of , , and .

The first two lengths can be found by subtracting known lengths:

The last two segments are hypotenuses of right triangles, and their lengths can be calculated using the Pythagorean Theorem:

is the hypotenuse of a triangle with legs ; it measures

is the hypotenuse of a triangle with legs ; it measures

Add the four sidelengths:

### Example Question #7 : Pythagorean Theorem

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Explanation:

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

We can set up a proportion statement by comparing the large triangle to the smaller of the two in which it is divided. The sides compared are the hypotenuse and the longer side:

Solve for :

### Example Question #8 : Pythagorean Theorem

A right triangle has one leg of length 42 inches; its hypotenuse has length 70 inches.  What is the area of this triangle?

Explanation:

The Pythagorean Theorem can be used to derive the length of the second leg:

inches.

Use the area formula for a triangle, with the legs as the base and height:

square inches.

### Example Question #9 : Pythagorean Theorem

The area of a right triangle is 136 square inches; one of its legs measures 8 inches. What is the length of its hypotenuse? (If not exact, give the answer to the nearest tenth of an inch.)

Explanation:

The area of a triangle is calcuated using the formula

In a right triangle, the bases can be used for base and height, so solve for  by substitution:

The legs measure 8 and 34 inches, respectively; use the Pythagorean Theorem to find the length of the hypotenuse:

inches

### Example Question #10 : Pythagorean Theorem

Refer to the above diagram. Which of the following expressions gives the length of  ?