8-2 The Pythagorean Theorem and Its Converse The student will be able to: 1.Use the Pythagorean Theorem. 2.Use the Converse of the Pythagorean Theorem.

The Pythagorean Theorem Pythagorean Theorem – 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 ΔABC is a right triangle with right angle C, then a 2 + b 2 = c 2. (Hint: c is always opposite the right angle.) Find the missing measures using the Pythagorean Theorem. a 2 + b 2 = c x = x = 625 x 2 = 141 |

Pythagorean triple – is a set of three nonzero whole numbers a, b, and c, such that a 2 +b 2 = c 2. One common Pythagorean triple is 3, 4, 5. The ratio of the sides of a right triangle is 3:4:5. The short leg is divisible by 3. The long leg is divisible by 4 and the hypotenuse is divisible by 5.

Example 1: Use the Pythagorean triple to find x. Explain your reasoning. 1 st – Do the two known measures have a GCF? Yes.4 2 nd – What are the ratios? 5, 12, 13Yes. 4 th – Since the Pythagorean triple is 5, 12, 13. The missing side (hypotenuse) is rd – Is there a common Pythagorean triple using 5 and 12? x = 52 Short leg Long leg Hypotenuse

Example 2: Real-world situation A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder? 1 st – Draw a picture. 20 ft 16 ft x 2 nd – Do the two known measures have a GCF? Yes.4 3 rd – What are the ratios? 3, 4, 5Yes. 5 th – Since the Pythagorean triple is 3, 4, 5. The missing side (short leg) is 3. 4 th – Is there a common Pythagorean triple using 4 and 5? x = 12 Short leg Long leg Hypotenuse

You Try It: 1. Use a Pythagorean triple to find x. Explain your reasoning. 2. According to your company’s safety regulations, the distance from the base of a ladder to a wall that it leans against should be at least one fourth of the ladder’s total length. You are given a 20-foot ladder to place against a wall at a job site. If you follow the company’s safety regulations, what is the maximum distance x up the wall the ladder will reach, to the nearest tenth ft Short leg - xLong leg = 24 Hypotenuse = 26 GCF - 2 x = 2(5) x = 10 a 2 + b 2 = c x 2 = x 2 = 400 x 2 = 375 x = 19.4 ft

The Converse of the Pythagorean Theorem if a 2 + b 2 = c 2, then ΔABC is a right triangle. The Converse of the Pythagorean Theorem – If the sum of the squares of the lengths of the shortest sides of a triangle is equal to the square of the length of the longest side of a triangle, then the triangle is a right triangle.

The Pythagorean Inequality Theorems If c 2 < a 2 + b 2, then ΔABC is an acute triangle. Pythagorean Inequality Theorem – If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides of a triangle, then the triangle is an acute triangle. If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides of a triangle, then the triangle is an acute triangle. if c 2 > a 2 + b 2, then ΔABC is an obtuse triangle.

Example 3: Determine whether each set of numbers can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. 11, 60, > 64 1 st – Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 71 > 64 true > > 60 true > > 11 true 2 nd –Classify the triangle by comparing the square of the longest side to the sum of the squares of the other two sides ? ? > 3721 This is an obtuse triangle. This is a triangle. c 2 ? a 2 + b 2