Sect. 9.2 & 9.3 The Pythagorean Theorem Goal 1 Proving the Pythagorean Theorem Goal 2 Using the Pythagorean Theorem

PROVING THE PYTHAGOREAN THEOREM The Pythagorean Theorem is one of the most famous theorems in mathematics. The relationship it describes has been known for thousands of years.

PROVING THE PYTHAGOREAN THEOREM THEOREM 9.4 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. b a c c 2 = a 2 + b 2

PROVING THE PYTHAGOREAN THEOREM Thus the sum of the squares on the smaller two sides equals the square on the biggest side.

PROVING THE PYTHAGOREAN THEOREM Example 1 Find the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple. 12 x 5 Because the side lengths 5, 12, and 13 are integers, they form a Pythagorean triple.

USING THE PYTHAGOREAN THEOREM A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 = a 2 + b 2. For example, the integers 3, 4, and 5 form a Pythagorean triple because 5 2 = 32 + 4 2.

PROVING THE PYTHAGOREAN THEOREM The most common Pythagorean Triples are: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25                     REMEMBER: The Pythagorean Theorem ONLY works in Right Triangles!

USING THE PYTHAGOREAN THEOREM Example 2 Many right triangles have side lengths that do not form a Pythagorean triple. x 14 7 Find the length of the leg of the right triangle. SOLUTION

USING THE PYTHAGOREAN THEOREM Example 3 Find x.

Finding the Area of a Triangle USING THE PYTHAGOREAN THEOREM Finding the Area of a Triangle Since the triangle is isosceles, it can be divided into two congruent right triangles

Find the missing side of the triangle and the Area. USING THE PYTHAGOREAN THEOREM Example 4 Find the missing side of the triangle and the Area.

Find the area of an equilateral triangle with perimeter 18 cm. USING THE PYTHAGOREAN THEOREM Example 5 Find the area of an equilateral triangle with perimeter 18 cm.

The Pythagorean Theorem and Baseball USING THE PYTHAGOREAN THEOREM Example 6 The Pythagorean Theorem and Baseball You've just picked up a ground ball at first base, and you see the other team's player running towards third base. How far do you have to throw the ball to get it from first base to third base, and throw the runner out?                                                                                                                  You need to throw the ball 127.3 feet to get it from first base to third base.

USING THE PYTHAGOREAN THEOREM Example 7 A ramp was constructed to load a truck.  If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp?                         

Theorem 9.5 Converse of the Pythagorean Theorem USING THE PYTHAGOREAN THEOREM Theorem 9.5 Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2 the triangle is a right triangle

Tests for Acute, Obtuse or Right Triangles: USING THE PYTHAGOREAN THEOREM Tests for Acute, Obtuse or Right Triangles: In triangle ABC, if c is the longest side of the triangle, then: Relations of sides    Type of Triangle              Acute Right Obtuse c2 < a2 + b2 c2 = a2 + b2 c2 > a2 + b2 Remember: The sum of the lengths of any two sides of a triangle must be greater than the 3rd side.

USING THE PYTHAGOREAN THEOREM Example 8 Decide whether the set of numbers can represent the sides of a triangle. If they can classify as Right, Acute, or Obtuse. a) 8, 18, 24 b) 3.2, 4.8, 5.1 c) 12.3, 16.4, 20.5 d) 8, 40, 41 e) 12, 16, 30

Homework 9.2 8-30 even 9.3 8-24 even