1. Lessons 9.1 – 9.2 The Pythagorean Theorem & Its Converse
HW: Lesson 9.1 / 1-16 evens and
Lesson 9.2/1-16 evens

2. Essential Understanding
Use the the Pythagorean Theorem to solve problems.
Use the Converse of the Pythagorean Theorem to solve problems.
Use side lengths to classify triangles by their angle measures.

3. If You Have A Right Triangle,
Pythagorean Theorem
If You Have A Right Triangle,
Then c²=a² + b² c a b

4. History of the theorem Pythagoras of Samos was a Greek
philosopher responsible for many important
developments in mathematics!
But rumour has it Pythagoras’ Theorem was
known to the Babylonians some 1000 years
before Pythagoras.
However we all believe he was the
first person to prove the theorem and
that is why the theorem takes his name.

5. The Pythagorean Theorem as some students see it.c b
c2 = a2 + b2

6. A better way c2 a2 c a
c2=a2+b2 b b2

7. c2=a2+b2 PYTHAGOREAN THEOREM Applies to Right Triangles Only!
hypotenuse c
leg a b
leg
c2=a2+b2

8. Find the missing side of the right triangle in
the 1 centimeter grid below. x 6 8

9. Find the missing side of the right triangle in
the 1 centimeter grid below. 12 5 x

10. Find the missing side of the right triangle in
the 1 centimeter grid below. 4 x 7

11. Find the length of the diagonal for a rectangle
that measures 3 inches by 4 inches. x
3 in.
4 in.

12. Find the Hypotenuse 1) To find the hypotenuse, solve for c.
2) a = 3m, b = 4m, find c.

13. Find a leg
You will not always solve for the hypotenuse (c). Sometimes you will have to find a leg (a or b).
Example:

14. To find a leg, solve for a or b.1)
2) b = 30ft, c = 34ft
34 m
30 m a

15. Pythagorean Theorem
c ≈ miles

16. Pythagoras Questions 1 2 x 3 cm 4 cm Pythagorean triple x 5 cm 12 cm
Pythagorean triple
5 cm
12 cm x 2
Pythagorean triple

17. x ≈ 6.32 cm x ≈ 21.11 cm Pythagoras Questions: Finding a leg measure 3
x m
9 m
11m 3
x ≈ 6.32 cm
Another method for finding a leg measure
11 cm
x cm
23.8 cm 4
x ≈ cm

18. Applications of Pythagoras
Find the diagonal of the rectangle
6 cm
9.3 cm 1 d
d = cm

19. x ≈ 6.51 cm Perimeter = 2(6.51+4.3) ≈ 21.62 cm 2
A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm.
Find its perimeter. 2
7.8 cm
4.3 cm
x cm
x ≈ 6.51 cm
Perimeter = 2( ) ≈ cm
therefore

20. Then You Have A Right Triangle
The Converse Of The
Pythagorean Theorem
If c² =a² + b²,
Then You Have A Right Triangle c a b

21. Using the Converse The Converse of the Pythagorean Theorem is True.
Remember “Converse” means “Reverse.”

22. Converse of the Pythagorean Theorem
If c2 = a2 + b2 , then the triangle with sides a, b, and c is a right triangle.
If a, b, and c are integers that satisfy the equation c2 = a2 + b2 =, then a, b, and c are known as Pythagorean triples.

23. Do These Lengths Form Right Triangles?
i.e. do they work in the Pythagorean Theorem?
5, 6, , 8, 10
10² __5² + 6²
100___
100≠ 61 NO
10²___6² + 8²
100___
100 = 100
YES

24. Example of the Converse
Determine whether a triangle with lengths 7, 11, and 12 form a right triangle.
**The hypotenuse is the longest length.
This is not a right triangle.

25. A Pythagorean Triple Is Any 3 Integers That Form A Right Triangle
3, 4, 5
Multiples Family
6,8,10
30,40,50
15,20,25
5, 12, 13
Multiples Family
10,24,26
25,60,65
35,84,91
Multiples of Pythagorean Triples are also Pythagorean Triples.

26. Example of the Converse
Determine whether a triangle with lengths 12, 20, and 16 form a right triangle.
This is a right triangle. A set of integers such
as 12, 16, and 20 is a Pythagorean triple.

27. Converse Examples Determine whether 4, 5, 6 is a Pythagorean triple.
15, 8, and 17 is a Pythagorean triple.
4, 5, and 6 is not a Pythagorean triple.
15, 8, and 17 is a Pythagorean triple.

28. Verifying Right Triangles? ?
The triangle is a right triangle.
Note: squaring a square root!!

29. Verifying Right Triangles? ? ?
The triangle is NOT a right triangle.
Note: squaring an integer & square root!!

30. What Kind of Triangle?? What Kind Of Triangle ? c² ?? a² + b²
You can use the Converse of the Pythagorean Theorem to verify that a given triangle is a right triangle or obtuse or acute.
What Kind Of Triangle ?
c² ?? a² + b²

31. If the c² = a² + b² , then right If the c² > a² + b² then obtuse
Triangle Inequality
What Kind Of Triangle ?
c² ?? a² + b²
If the c² = a² + b² , then right
If the c² > a² + b² then obtuse
If the c² < a² + b², then acute
The converse of the Pythagorean Theorem can be used to categorize triangles.

32. The converse of the Pythagorean Theorem can be used to categorize triangles.
If c2 = a2 + b2, then triangle ABC is a right triangle.
If c2 > a2 + b2, then triangle ABC is an obtuse triangle.
If c2 < a2 + b, then triangle ABC is an acute triangle.

33. Triangle Inequality The triangle is obtuse 38, 77, 86 c2 ? a2 + b2
862 ?
7396 ?
7396 > 7373
The triangle is obtuse

34. Triangle Inequality The triangle is acute 10.5, 36.5, 37.5
c2 ? a2 + b2
37.52 ? ?
<
The triangle is acute

35. 4,7,9
9²__4² + 7²
81__
81 > 65
OBTUSE
greater

36. 5,5,7
7² __5² + 5² __
49 < 50
ACUTE
Less than

38. 25=9 + 16 52=32+ 42 25 9 16 A Pythagorean Triple 3, 4, 5
In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 25 9 16
52=32+ 42
25=9 + 16

39. 169=25 + 144 169 A 2nd Pythagorean Triple 5, 12, 13 25 144
In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
169
144 25
132 =
169=

40. A 3rd Pythagorean Triple
625
576 49
252 =
625= 7 24 25
A 3rd Pythagorean Triple
7, 24, 25

42. Building a foundation
Construction: You use four stakes and string to mark the foundation of a house. You want to make sure the foundation is rectangular.
a. A friend measures the four sides to be 30 feet, 30 feet, 72 feet, and 72 feet. He says these measurements prove that the foundation is rectangular. Is he correct?

43. Building a foundation
Solution: Your friend is not correct. The foundation could be a nonrectangular parallelogram, as shown below.

44. Building a foundation
b. You measure one of the diagonals to be 78 feet. Explain how you can use this measurement to tell whether the foundation will be rectangular.

45. Building a foundation
Because = 782, you can conclude that both the triangles are right triangles. The foundation is a parallelogram with two right angles, which implies that it is rectangular
Solution: The diagonal divides the foundation into two triangles. Compare the square of the length of the longest side with the sum of the squares of the shorter sides of one of these triangles.