Math: The Pythagorean Theorem Worksheet

Question 1

A right triangle has legs of length 3 units and 4 units. What is the length of the hypotenuse?

Accepted Answer

The hypotenuse is 5 units because 3 squared plus 4 squared equals 9 plus 16, which equals 25, and the square root of 25 is 5.

Question 2

A right triangle has legs of length 5 units and 12 units. Find the hypotenuse.

Accepted Answer

The hypotenuse is 13 units because 5 squared plus 12 squared equals 25 plus 144, which equals 169, and the square root of 169 is 13.

Question 3

A right triangle has a hypotenuse of 10 units and one leg of 6 units. Find the other leg.

Accepted Answer

The other leg is 8 units because 6 squared plus b squared equals 10 squared, so 36 plus b squared equals 100. Then b squared equals 64, and b equals 8.

Question 4

A right triangle has legs of 8 units and 15 units. What is the hypotenuse?

Accepted Answer

The hypotenuse is 17 units because 8 squared plus 15 squared equals 64 plus 225, which equals 289, and the square root of 289 is 17.

Question 5

A ladder leans against a wall. The foot of the ladder is 9 feet from the wall, and the ladder is 15 feet long. How high up the wall does the ladder reach?

Accepted Answer

The ladder reaches 12 feet up the wall because h squared plus 9 squared equals 15 squared. Then h squared plus 81 equals 225, so h squared equals 144 and h equals 12.

Question 6

A rectangle is 7 units wide and 24 units long. What is the length of its diagonal?

Accepted Answer

The diagonal is 25 units because the diagonal of a rectangle forms a right triangle with the side lengths. Then 7 squared plus 24 squared equals 49 plus 576, which equals 625, and the square root of 625 is 25.

Question 7

A right triangle has a hypotenuse of 25 units and one leg of 7 units. Find the other leg.

Accepted Answer

The other leg is 24 units because 7 squared plus b squared equals 25 squared. Then 49 plus b squared equals 625, so b squared equals 576 and b equals 24.

Question 8

Determine whether a triangle with side lengths 6, 8, and 10 is a right triangle. Explain your reasoning.

Accepted Answer

Yes, it is a right triangle because 6 squared plus 8 squared equals 36 plus 64, which equals 100, and 10 squared also equals 100. Since the equation is true, the triangle is a right triangle.

Question 9

Determine whether a triangle with side lengths 7, 10, and 12 is a right triangle. Explain your reasoning.

Accepted Answer

No, it is not a right triangle because 7 squared plus 10 squared equals 49 plus 100, which equals 149, but 12 squared equals 144. Since the values are not equal, the triangle is not a right triangle.

Question 10

A square has side length 9 units. What is the length of its diagonal? Give an exact answer in simplest radical form.

Accepted Answer

The diagonal is 9 times the square root of 2 units because 9 squared plus 9 squared equals d squared, so 81 plus 81 equals 162, and the square root of 162 simplifies to 9 times the square root of 2.

Question 11

A right triangle has legs 10 units and 10 units. What is the hypotenuse? Give an exact answer in simplest radical form.

Accepted Answer

The hypotenuse is 10 times the square root of 2 units because 10 squared plus 10 squared equals c squared, so 100 plus 100 equals 200, and the square root of 200 simplifies to 10 times the square root of 2.

Question 12

A park has a rectangular field that is 20 meters by 21 meters. What is the length of the field's diagonal?

Accepted Answer

The diagonal is 29 meters because 20 squared plus 21 squared equals 400 plus 441, which equals 841, and the square root of 841 is 29.

Question 13

A right triangle has a hypotenuse of 13 units and one leg of 5 units. Find the other leg.

Accepted Answer

The other leg is 12 units because 5 squared plus b squared equals 13 squared. Then 25 plus b squared equals 169, so b squared equals 144 and b equals 12.

Question 14

A triangle has side lengths 9, 12, and 15. Is it a right triangle? Explain.

Accepted Answer

Yes, it is a right triangle because 9 squared plus 12 squared equals 81 plus 144, which equals 225, and 15 squared equals 225. Since both sides are equal, the triangle is a right triangle.

Question 15

A television screen measures 18 inches wide and 24 inches tall. What is the length of the screen's diagonal?

Accepted Answer

The diagonal is 30 inches because 18 squared plus 24 squared equals 324 plus 576, which equals 900, and the square root of 900 is 30.

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Math: The Pythagorean Theorem

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