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

2. 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.

3. 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

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

5. 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.

6. 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 =

7. 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!

8. 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

9. USING THE PYTHAGOREAN THEOREM
Example 3
Find x.

10. 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

11. 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.

12. 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.

13. 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 feet to get it from first base to third base.

14. 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?
                        

15. 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

16. 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.

17. 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

18. Homework
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even