1. Objectives/Assignment
Prove the Pythagorean Theorem
Use the Pythagorean Theorem to solve real-life problems such as determining how far a ladder will reach.
Assignment: pp #1-31 all
Assignment due today: 9.1 pp #1-34 all

2. History Lesson
Around the 6th century BC, the Greek mathematician Pythagorus founded a school for the study of philosophy, mathematics and science. Many people believe that an early proof of the Pythagorean Theorem came from this school.
Today, the Pythagorean Theorem is one of the most famous theorems in geometry. Over 100 different proofs now exist.

3. Proving the Pythagorean Theorem
In this lesson, you will study one of the most famous theorems in mathematics—the Pythagorean Theorem. The relationship it describes has been known for thousands of years.

4. 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 legs.
c2 = a2 + b2

5. Given: In ∆ABC, BCA is a right angle.
Prove: c2 = a2 + b2
Proof:
Plan for proof: Draw altitude CD to the hypotenuse just like in Then apply Geometric Mean Theorem 9.3 which states that when the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

6. Proof Statements: Reasons: Perpendicular Postulate Geometric Mean Thm.
1. Draw a perpendicular from C to AB.
and
Reasons:
Perpendicular Postulate
Geometric Mean Thm.
Cross Product Property c a c b = = a e b f
3. ce = a2 and cf = b2
4. ce + cf = a2 + b2
4. Addition Property of =
5. c(e + f) = a2 + b2
5. Distributive Property
6. Segment Add. Postulate
6. e + f = c
7. c2 = a2 + b2
7. Substitution Property of =

7. Using the Pythagorean Theorem
A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2 For example, the integers 3, 4 and 5 form a Pythagorean Triple because 52 =

8. Ex. 1: Finding the length of the hypotenuse.
Find the length of the hypotenuse of the right triangle. Tell whether the sides lengths form a Pythagorean Triple.

9. Solution: Pythagorean Theorem x2 = 52 + 122 Substitute values.
(hypotenuse)2 = (leg)2 + (leg)2
x2 =
x2 =
x2 = 169
x = 13
Because the side lengths 5, 12 and 13 are integers, they form a Pythagorean Triple. Many right triangles have side lengths that do not form a Pythagorean Triple as shown next slide.
Pythagorean Theorem
Substitute values.
Multiply
Add
Find the positive square root.
Note: There are no negative square roots until you get to Algebra II and introduced to “imaginary numbers.”

10. Ex. 2: Finding the Length of a Leg
Find the length of the leg of the right triangle.

11. Solution:
(hypotenuse)2 = (leg)2 + (leg)2
142 = 72 + x2
196 = 49 + x2
147 = x2
√147 = x
√49 ∙ √3 = x
7√3 = x
Pythagorean Theorem
Substitute values.
Multiply
Subtract 49 from each side
Find the positive square root.
Use Product property
Simplify the radical.
In example 2, the side length was written as a radical in the simplest form. In real-life problems, it is often more convenient to use a calculator to write a decimal approximation of the side length. For instance, in Example 2, x = 7 ∙√3 ≈ 12.1

12. Ex. 3: Finding the area of a triangle
Find the area of the triangle to the nearest tenth of a meter.
You are given that the base of the triangle is 10 meters, but you do not know the height.
Because the triangle is isosceles, it can be divided into two congruent triangles with the given dimensions. Use the Pythagorean Theorem to find the value of h.

13. Solution: Now find the area of the original triangle. Steps:
(hypotenuse)2 = (leg)2 + (leg)2
72 = 52 + h2
49 = 25 + h2
24 = h2
√24 = h
Reason:
Pythagorean Theorem
Substitute values.
Multiply
Subtract 25 both sides
Find the positive square root.
Now find the area of the original triangle.

14. Area of a Triangle Area = ½ bh = ½ (10)(√24) ≈ 24.5 m2
The area of the triangle is about 24.5 m2

15. Ex. 4: Indirect Measurement
Support Beam: The skyscrapers shown on page 535 are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.

16. Each support beam forms the hypotenuse of a right triangle
Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).

17. Solution: Pythagorean Theorem x2 = (23.26)2 + (47.57)2
(hypotenuse)2 = (leg)2 + (leg)2
x2 = (23.26)2 + (47.57)2
x2 = √ (23.26)2 + (47.57)2
x ≈ 13
Pythagorean Theorem
Substitute values.
Multiply and find the positive square root.
Use a calculator to approximate.

18. Objectives/Assignment
Use the Converse of the Pythagorean Theorem to solve problems.
Use side lengths to classify triangles by their angle measures.
Assignment: pp #1-35
Assignment due today: 9.2
Reminder Quiz after this section on Monday.

19. Using the Converse
In Lesson 9.2, you learned that if a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs. The Converse of the Pythagorean Theorem is also true, as stated on the following slide.

20. Converse of the Pythagorean Theorem
If the square of the length of the longest side of the 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, then ∆ABC is a right triangle.

21. Note:
You can use the Converse of the Pythagorean Theorem to verify that a given triangle is a right triangle, as shown in Example 1.

22. Ex. 1: Verifying Right Triangles
The triangles on the slides that follow appear to be right triangles. Tell whether they are right triangles or not.
√113
4√95

23. Ex. 1a: Verifying Right Triangles
Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c2 = a2 + b2.
(√113)2 =
113 =
113 = 113 ✔
√113 ? ?
The triangle is a right triangle.

24. Ex. 1b: Verifying Right Triangles
c2 = a2 + b2.
(4√95)2 =
42 ∙ (√95)2 =
16 ∙ 95 =
1520 ≠ 1521 ✔
4√95 ? ? ?
The triangle is NOT a right triangle.

25. Classifying Triangles
Sometimes it is hard to tell from looking at a triangle whether it is obtuse or acute. The theorems on the following slides can help you tell.

26. Triangle Inequality
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, then the triangle is acute.
If c2 < a2 + b2, then ∆ABC is acute
c2 < a2 + b2

27. Triangle Inequality
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, then the triangle is obtuse.
If c2 > a2 + b2, then ∆ABC is obtuse
c2 > a2 + b2

28. Classifying Triangles
Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse.
38, 77, 86 b , 36.5, 37.5
You can use the Triangle Inequality to confirm that each set of numbers can represent the side lengths of a triangle. Compare the square o the length of the longest side with the sum of the squares of the two shorter sides.

29. Triangle Inequality to confirm
Reason:
Compare c2 with a2 + b2
Substitute values
Multiply
c2 is greater than a2 + b2
The triangle is obtuse
Statement:
c2 ? a2 + b2
862 ?
7396 ?
7395 > 7373

30. Triangle Inequality to confirm
Statement:
c2 ? a2 + b2
37.52 ? ?
<
Reason:
Compare c2 with a2 + b2
Substitute values
Multiply
c2 is less than a2 + b2
The triangle is acute

31. Ex. 3: 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?

32. Ex. 3: Building a foundation
Solution: Your friend is not correct. The foundation could be a nonrectangular parallelogram, as shown below.

33. Ex. 3: 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.

34. Ex. 3: 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.