the pythagorean theorem and its activity …tristanbates.wikispaces.com/file/view/geometry - unit...

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Unit 3 • Similarity, Right Triangles, and Trigonometry 235

My Notes

ACTIVITY

3.6The Pythagorean Theorem and ItsConverse Is That Right?SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Shared Reading, Summarize/Paraphrase/Retell

How did Pythagoras get a theorem named a! er him?Although many examples of the Pythagorean ! eorem were known

and used by the Babylonians, Chinese, Hindu and Egyptians well before Pythagoras was born (about 570 BCE), he is given credit for being the " rst to formally prove the theorem. Many others since Pythagoras’ time, including a young man named James Gar" eld who would go on to be President of the United States, have also o# ered formal proofs of the well known theorem.

Examine one proof of the Pythagorean $ eorem that is credited to Pythagoras himself.

Begin with a square having edges of length a + b. In the square, four right triangles with legs a and b have been drawn.

N OM

R

P

QS T

a b

b

a

ba

b a

1. Each of the four right triangles in the diagram above are congruent. What triangle congruence method justi" es this statement? Explain your answer.

2. Since the four right triangles are congruent, we know theirhypotenuses,

__ RN , __

TR , __

PT and __

NP , are congruent.

a. What reason can be used to justify this?

b. Label each hypotenuse in the diagram, c.

ACADEMIC VOCABULARY

THE PYTHAGOREAN THEOREMIn any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

If a and b are the lengths of the legs and c is the length of the hypotenuse then, c2 = a2 + b2.

235-242_SB_Geom_3-6_SE.indd 235235-242_SB_Geom_3-6_SE.indd 235 1/21/10 2:20:31 PM1/21/10 2:20:31 PM

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236 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

ACTIVITY 3.6continued Is That Right?Is That Right?

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Look for a Pattern, Quickwrite

3. !MNR is a right triangle and !MNR " !SRT.

a. What is the relationship between ∠MRN and ∠MNR?How do you know?

b. Use the congruence statement, ∠MNR " ∠SRT.What does this indicate about the relationship between ∠MRN and ∠SRT? Explain your reasoning.

c. What kind of angle is ∠NRT ? How do you know?

d. What are the measures of ∠RTP, ∠TPN and ∠PNR ? Justify your answer.

4. What special quadrilateral is formed by the four hypotenuses? Justify your answer.

The Pythagorean Theorem and Its Converse


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My Notes

ACTIVITY 3.6continuedIs That Right?Is That Right?

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations

5. It can be assumed from the diagram that the area of the large outside square is equal to the sum of the areas of the four triangles and quadri-lateral PNRT. Write an equation, in terms of a, b, and c that represents this statement.

6. Use algebraic properties to simplify both sides of the equation.

7. Solve the simpli! ed equation for c2.

You have now veri! ed algebraically, much as Pythagoras is thought to have done, " e Pythagorean " eorem and can use it to solve problems.

8. How high up a vertical wall will a 24 foot ladder reach if the foot of the ladder is placed 10 feet from the wall? Draw a sketch and show the calculations that support your answer.



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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Marking the Text, Think/Pair/Share, Self/Peer Revision

9. Find the area of a rectangular rug if the width of the rug is 13 feetand the diagonal measures 20 feet. Draw a sketch and show the calculations that support your answer.

! e Pythagorean ! eorem states that, 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. Is the converse of this statement true?

10. Write the Pythagorean ! eorem in if-then form.

11. Write the converse of the Pythagorean ! eorem in if-then form.

12. Can the converse of the Pythagorean ! eorem be proven? Assume you have !ABC where c2 = a2 + b2, as shown below. Complete the following to try to prove !ABC is a right triangle. Use right !DEF, with legs a and b and hypotenuse f.

B

CA

a

b

c

E

FD

a

b

f



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SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Close Reading, Group Presentation, Quickwrite

a. It is known that f 2 = a2 + b2. Give a reason for this statement.

b. It was assumed that in !ABC, c2 = a2 + b2. So, the statementc = f can be made. Why is this true?

c. !ABC " !DEF by what reason?

d. ∠C is a right angle. Give a reason for this statement.

e. !ABC is a right triangle. What reason justi! es this statement?

13. You examined the converse of the Pythagorean " eorem. Now, take a look at the inverse.

a. Write the inverse of the Pythagorean " eorem in if-then form.

b. Is the inverse a true statement? Why or why not?



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My Notes


SUGGESTED LEARNING STRATEGIES: Use Manipulatives

Since you have shown the Converse of the Pythagorean ! eorem is true, a little more exploration follows.

14. Use each of the following sets of triangle side lengths to build triangles using the manipulatives (straws) provided by your teacher.

Step 1: Cut manipulatives into 5 cm, 6 cm, 12 cm, 13 cm, and 15 cm lengths.

Step 2: Build each triangle on centimeter grid paper.

Step 3: Identify each triangle as right, acute or obtuse.

Step 4: Complete the table.

Triangle side lengths Type of triangle c 2 a2 + b2

5, 12, 13

6, 6, 12

5, 6, 12

5, 12, 15

5, 12, 12

6, 12, 13

6, 12, 15



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My Notes


SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Look for a Pattern

15. What does your work in Item 14 suggest about the relationship between a2, b2, c 2 and the type of triangle?

16. Use the Converse of the Pythagorean ! eorem to determine whether each of the following sets of side lengths forms a right triangle. If a right triangle is not possible, tell whether an acute or obtuse triangle can be formed. Show the method you use to determine your answers.

a. 12, 34, 37

b. 6 __ 7 , 8 __ 7 , 10 ___ 7

c. 20, √___

42 , 21



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Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper or on grid paper. Show your work.

1. If a television screen is a rectangle with a 53 inch diagonal and a width of 45 inches, what is the height of the screen?

2. A standard baseball diamond is a square 90 feet on each side. Find the distance of a throw made from the catcher 3 feet behind home plate in an attempt to throw out a runner trying to steal second base. Round to the nearest whole number.

a. 93 feet b. 124 feet

c. 130 feet d. 183 feet

3. Tell whether a triangle can be formed having the following side lengths. If a triangle can be formed tell whether it is right, acute or obtuse.

a. 4, 6, 8 b. √__

8 , √__

8 , √___

16

4. MATHEMATICAL R E F L E C T I O N

! e Pythagorean ! eorem was thought of by the early

Greeks as the following: ! e area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the legs.

Draw a diagram to illustrate this statement. Explain how your diagram illustrates the Pythagorean ! eorem.


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge

Another way to prove the Pythagorean theorem is by using triangle similarity. In right triangle ABC below, an altitude is drawn to hypotenuse AB, forming two right triangles that are similar to triangle ABC.

Corresponding sides of similar triangles are in proportion, so you can write these proportions involving sides of the triangles.

b __ x = c __ b a ____ c – x = c __ a

17. Use the proportions above and algebra to prove a2 + b2 = c2.

A B

C

b

c

ah

x c – x


the pythagorean theorem and its activity …tristanbates.wikispaces.com/file/view/geometry - unit...

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