Section 1 – Apply the Pythagorean Theorem Unit 6 – Right Triangles Section 1 – Apply the Pythagorean Theorem

Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which option would you choose?

Find the length of the hypotenuse of the right triangle. Example 1 Find the length of the hypotenuse of the right triangle. SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem x2 = 62 + 82 Substitute. x2 = 36 + 64 Multiply. x2 = 100 Add. x = 10 Find the positive square root.

Example 2 Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form. SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem 52 = 32 + x2 Substitute. 25= 9 + x2 Multiply. x2 = 16 Subtract 9 from both sides x = 4 Find the positive square root. Leg;4

GUIDED PRACTICE Example 3 Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form. SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem x2 = 62 + 42 Substitute. x2 = 36 + 16 Multiply. x2 = 52 Add. x = 2 13 Find the positive square root. hypotenuse; 2 13

Example 4 SOLUTION = +

EXAMPLE 2 Example 4 SOLUTION 162 = 42 + x2 256 = 16 + x2 240 = x2 Substitute. 256 = 16 + x2 Multiply. 240 = x2 Subtract 16 from each side. 240 = x Find positive square root. 15.491 ≈ x Approximate with a calculator. ANSWER The ladder is resting against the house at about 15.5 feet above the ground. The correct answer is D.

GUIDED PRACTICE Example 5 The top of a ladder rests against a wall, 23 feet above the ground. The base of the ladder is 6 feet away from the wall. What is the length of the ladder? SOLUTION = +

GUIDED PRACTICE Example 5 x2 = (6)2 + (23)2 x2 = 36 + 529 x2 = 565 Substitute. x2 = 36 + 529 Multiply. x2 = 565 Add. x = 23.77 Approximate with a calculator. about 23.8 ft ANSWER

GUIDED PRACTICE Example 7 The Pythagorean Theorem is only true for what type of triangle? right triangle ANSWER

EXAMPLE 8 Find the area of an isosceles triangle Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. SOLUTION STEP 1 Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown.

Find the area of an isosceles triangle EXAMPLE 8 Find the area of an isosceles triangle Use the Pythagorean Theorem to find the height of the triangle. STEP 2 c2 = a2 + b2 Pythagorean Theorem 132 = 52 + h2 Substitute. 169 = 25 + h2 Multiply. 144 = h2 Subtract 25 from each side. 12 = h Find the positive square root.

EXAMPLE 8 Find the area of an isosceles triangle STEP 3 Find the area. 1 2 (base) (height) = (10) (12) = 60 m2 1 2 Area = ANSWER The area of the triangle is 60 square meters.

Example 9 Find the area of the triangle. SOLUTION To find area of a triangle, first altitude has to be ascertained. STEP 1 By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles.

GUIDED PRACTICE STEP 2 c2 = a2 + b2 182 = 152 + h2 324 = 225 + h2 Pythagorean Theorem 182 = 152 + h2 Substitute. 324 = 225 + h2 Multiply. 99 = h2 Subtract 225 from each side. 9.95 = h Find the positive square root.

GUIDED PRACTICE STEP 3 Find the area. 1 2 (base) (height) = 30 9.95 = 149.25 ft2 1 2 Area = ANSWER The area of the triangle is 149.25 ft2.

Find the area of the triangle. GUIDED PRACTICE Example 10 Find the area of the triangle. SOLUTION c2 = a2 + b2 Pythagorean Theorem 262 = 102 + h2 Substitute. 676 = 100 + h2 Multiply. 576 = h2 Subtract 100 from each side. 24 = h Find the positive square root.

GUIDED PRACTICE 1 2 1 2 Area = (base) (height) = (20) (24) = 240 m2 ANSWER The area of the triangle is 240 square meters.

Pythagorean Triples A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation .

The Most Common Pythagorean Triples and Their Multiples

Example 11 Find the length of a hypotenuse of a right using two methods. 32 24 x