**Using the pythagorean theorem :**

We can use the Pythagorean Theorem to find the length of a side of a right triangle when we know the lengths of the other two sides.

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

If a and b are legs and c is the hypotenuse, then

a² + b² = c²

**Problem 1 : **

In the right triangle given below, find the length of the missing side using Pythagorean theorem.

**Solution :**

**Step 1 : **

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b² = c²

**Step 2 :**

Substitute the given measures.

7² + 24² = c²

**Step 3 :**

Solve the equation for c.

7² + 24² = c²

Simplify.

49 + 576 = c²

625 = c²

Write 625 as a perfect square (625 = 25²).

25² = c²

Get rid of the square on both sides.

25 = c

Hence, the length of the hypotenuse is 25 inches.

**Problem 2 : **

In the right triangle given below, find the length of the missing side using Pythagorean theorem.

**Solution :**

**Step 1 :**

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b² = c²

**Step 2 :**

Substitute the given measures.

a² + 12² = 15²

**Step 3 :**

Solve the equation for c.

Simplify.

a² + 144 = 225

Subtract 144 from both sides.

a² + 144 - 144 = 225 - 144

a² = 81

Write 81 as a perfect square (81 = 9²).

a² = 9²

Get rid of the square on both sides.

a = 9

Hence, the length of the leg is 9 centimeters.

**Problem 3 : **

In the right triangle given below, find the length of the missing side using Pythagorean theorem.

**Solution :**

**Step 1 :**

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b² = c²

**Step 2 :**

Substitute the given measures.

30² + 40² = c²

**Step 3 :**

Solve the equation for c.

30² + 40² = c²

Simplify.

900 + 1600 = c²

2500 = c²

Write 2500 as a perfect square (2500 = 50²).

50² = c²

Get rid of the square on both sides.

50 = c

Hence, the length of the hypotenuse is 50 ft.

**Problem 4 : **

In the right triangle given below, find the length of the missing side using Pythagorean theorem.

**Solution :**

**Step 1 :**

If a and b are legs and c is the hypotenuse, write Pythagorean for the above right triangle

a² + b² = c²

**Step 2 :**

Substitute the given measures.

a² + 40² = 41²

**Step 3 :**

Solve the equation for c.

Simplify.

a² + 1600 = 1681

Subtract 1600 from both sides.

a² + 1600 - 1600 = 1681 - 1600

a² = 81

Write 81 as a perfect square (81 = 9²).

a² = 9²

Get rid of the square on both sides.

a = 9

Hence, the length of the leg is 9 inches.

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