IM 8 Ch 9.2.7 Does It Always Work -...

IM 8 Ch 9.2.7 Does It Always Work

CPM Materials modified by Mr. Deyo

Is the triangle a right triangle?

Who is Pythagoras?

Common Core Standard: 8.G.6


By the end of the period, I will explain a proof of the Pythagorean Theorem and its converse.

I will demonstrate this by completing Four‑Square notes and by solving problems in a pair/group activity.

Learning TargetTitle: IM8 ‑ Ch. 9.2.7 Does It Always Work? Date:


Home Work: Sec. 9.2.7Desc. Date Due

Review & Preview

4 Problems 9‑149, 9‑150, 9‑151, 9‑154


Vocabulary1) right triangle

2) leg(s)(a & b)

3) hypotenuse (c)

4) Pythagorean Theorema2 + b2 = c2


9.2.7 Does It Always Work? You have seen that you can find the missing side of a right triangle using the Pythagorean Theorem. To show that it is always true, no matter how long the sides are, it must be proven. There are over 100 different ways to prove this important relationship. Today you will look at two ways to prove the Pythagorean Theorem. As you may remember, the Pythagorean Theorem states that in a right triangle with sides a and b, and hypotenuse c, a2 + b2 = c2.9145 Obtain the Lesson 9.2.7A Resource Page from your teacher. For additional support, watch Pythagorean Proof Video

b

ac

b

a c

http://www.cpm.org/pdfs/stuRes/CC3/chapter_09/CC3%20Lesson%209.2.7A%20RP.pdf

http://www.cpm.org/media/Pythagorean_Theorem.m4v




9145a,b) Start by cutting out four copies of the right triangle with legs labeled a, b, and hypotenuse labeled c units.a) First, arrange the triangles to look like the diagram at right. Draw this diagram on your paper. Explain why the area of the unshaded region is c2.b) Will moving the triangles within the bold outer square change the total unshaded square?

b

ac

b

a c






9145) c) Move the shaded triangles to match the diagram at right. In this arrangement, tell why the total area that is unshaded is a2 + b2.d) Write an equation that relates the unshaded region in part (a) to the unshaded region in part (b).

b

ac

b

a c






9146. Here is another proof of the Pythagorean Theorem for you to try. Obtain the Lesson 9.2.7B Resource Page from your teacher. a) Start with two squares as shown here. What is the total area?

c) Cut out the shaded triangles shown in the diagram at right. Then work with your team to determine how to arrange the shaded triangles and the unshaded portion of the original figure to create a square. What is the area of the square? How do you know?

a

b

ca

bc

b) Use a ruler or another piece of paper to place a mark of length b on the bottom left side of the larger square. Then draw the dotted lines as shown in the diagram at right to create two right triangles. How do you know that the legs of both triangles are legs a and b? Label each hypotenuse c.

d) How does what you have done in this problem prove the Pythagorean Theorem?

IM 8 Ch 9.2.7 Does It Always Workhttp://www.cpm.org/pdfs/stuRes/CC3/chapter_09/CC3%20Lesson%209.2.1A%20RP.pdf

9.2.1A#951

147b)



9147. The converse of a theorem reverses the evidence and the conclusion. The Pythagorean Theorem states that in a right triangle with legs of a and b, and hypotenuse c, that a2 + b2 = c2.a) State the converse of the Pythagorean Theorem.

b) Look back at your work from problem 951. What can you conclude about a triangle if a2 + b2 = c2?

c) Why is this not a proof of the converse of the Pythagorean Theorem?


9148. Graph the points A(−2, 2) and B(1, −2). Then find the distance between them by creating a right triangle (like a slope triangle) and computing the length of the hypotenuse.


9149. Use a graph to find the distance between the points C(−4, −1) and D(4,1). http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/

chapter/Ch9/lesson/9.2.7/problem/9149https://www.desmos.com/calculator/4h2mt3s7qy

http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch9/lesson/9.2.7/problem/9-149

https://www.desmos.com/calculator/4h2mt3s7qy


9150. Jack has a tree in his backyard that he wants to cut down to ground level. He needs to know how tall the tree is, because when he cuts it, it will fall toward his fence. Jack measured the tree’s shadow, and it measured 20 feet long. At the same time, Jack’s shadow was 12 feet long. Jack is 5 feet tall.

a) How tall is the tree?

http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch9/lesson/9.2.7/problem/9150

Tree Jack

b) Will the tree hit the fence if the fence is 9 feet away?



9151. Examine the diagrams below. What is the geometric relationship between the labeled angles? What is the relationship of their measures? After you determine the relationship of their measures, use the relationship to write an equation and solve for x.

a) b)




9152. On graph paper, graph the system of equations below. Graph

y = x + 1

What's the solution of the system? ( , )If there is no solution, explain why not.

https://www.desmos.com/calculator/8ltjjowa9q http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch9/lesson/9.2.7/problem/9152

25

y = x 225

https://www.desmos.com/calculator/8ltjjowa9q



9153. A principal made the histogram at right to analyze how many years teachers had been teaching at her school.

c) What is the probability that a teacher on the staff has been there for fewer than 5 years?

b) If the principal randomly chose one teacher to represent the school at a conference, what is the probability that the teacher would have been teaching at the school for more than 10 years? Write the probability in two different ways.

a) How many teachers work at her school?




9154a,b. Simplify each expression.

a) b)


12 5

7 10

94

13( )



9154c,d. Simplify each expression.

c) d*)


35

16

95

10 3

( )( )


IM 8 Ch 9.2.7 Does It Always Work -...

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