(8) Inquiry Lab - Proofs About the Pythagorean Theorem
- 1. HOW can you prove the Pythagorean Theorem and its converse? Course 3, Inquiry Lab after Lesson 5-5 Geometry
- 2. • 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. 7 Look for and make use of structure. Course 3, Inquiry Lab after Lesson 5-5 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Geometry
- 3. Course 3, Inquiry Lab after Lesson 5-5 Activity 1 continued Geometry The Pythagorean Theorem is named after the famous Greek mathematician Pythagoras who lived around 500 B.C. The properties of the theorem, however, were known by the ancient Egyptians, Babylonians, and Chinese. The following geometric proof is similar to a visual proof shown in a Chinese document written between 500 B.C. and 200 B.C. Draw and cut out 8 copies of a right triangle. Label leach pair of legs a and b, and each hypotenuse c.
- 4. Course 3, Inquiry Lab after Lesson 5-5 Activity1 continued Geometry On a separate piece of paper arrange four of the triangles in a square as shown. Trace the figure formed by the hypotenuses. The length of each side of the large square is a + b, so the area of the large square is 2 ( ) .a b Is the figure formed by the hypotenuses a square? Explain. _________________________________________________________ _________________________________________________________ _________________________________________________________ Write an expression for the area of the inside square. _______________
- 5. Course 3, Inquiry Lab after Lesson 5-5 Activity 1 continued Geometry On the same paper, arrange the remaining triangles as shown. Draw the two figures shown by the dashed lines. The length of each side of the large square is a + b, so the area of the large square is 2 ( ) .a b Are the two figures represented by dashed lines squares? Explain. _______________________________________________________ _______________________________________________________ Write an expression for the area of the small square. Write an expression for the area of the large square.
- 6. Course 3, Inquiry Lab after Lesson 5-5 Geometry Activity 1 continued Since the area of each of the two composite figures you created is , the areas are equal. Use the space provided to draw each figure from Step 2 and Step 3. Place an equal sign between the two figures to show the two areas are equal. 2 ( )a b
- 7. Course 3, Inquiry Lab after Lesson 5-5 Geometry Remove the triangles from each side. Use the space provided to draw the remaining figures. What property justifies removing the triangles from each side of the equation? ___________________________________________________ Write an algebraic equation that represent the relationship between the figures shown in Step 5. ________________________________________ Summarize the relationship among the sides of a right triangle measuring a units, b units, and c units. _____________________________________ ___________________________________________________________
- 8. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. 2 2 2 a b c Given: . Prove: is a right triangle. such thatABC 2 2 2 a b c ABC Complete the proof with the correct reasons justifying each statement. _____________________________ Statements Reasons a. Draw a right triangle DEF so that is b units long. Label as d. DF FE
- 9. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry Statements Reasons The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. such thatABC 2 2 2 a b c ABC 2 2 2 a b c b. Write an equation that describes the relationship between the side lengths of . State the theorem that allows you to make that statement. DEF ______________________ ______________________ ______________________
- 10. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. 2 2 2 a b c such thatABC 2 2 2 a b c ABC Statements Reasons 2 2 2 a b c Givenc.
- 11. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. 2 2 2 a b c such thatABC 2 2 2 a b c ABC Statements Reasons d. If and then . 2 2 2 a b c 2 2 2 a b d 2 2 d c ______________________ ______________________ ______________________
- 12. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. Statements Reasons 2 2 2 a b c such thatABC 2 2 2 a b c ABC e. If , then d = c.2 2 d c ______________________ ______________________ ______________________
- 13. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. Statements Reasons ______________________ ______________________ ______________________ 2 2 2 a b c such thatABC 2 2 2 a b c ABC f. If d = c then FE = AB.
- 14. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. Statements Reasons 2 2 2 a b c such thatABC 2 2 2 a b c ABC g. If AC = FD, CB = DE, and AB = FE, the two triangles are the same shape and size. If three sides of a triangle are the same length as the corresponding sides of another triangle, the triangles are the same shape and size.
- 15. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. Statements Reasons Corresponding parts of the triangles with the same size and shape have the same measure. 2 2 2 a b c such thatABC 2 2 2 a b c ABC h. m C m D
- 16. Course 3, Inquiry Lab after Lesson 5-5 Activity 2 continued Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. Statements Reasons 2 2 2 a b c such thatABC 2 2 2 a b c ABC i. is a right angle.C ______________________ ______________________ ______________________
- 17. Course 3, Inquiry Lab after Lesson 5-5 Geometry The converse of the Pythagorean Theorem states if a triangle has side lengths, a, b, and c such that , then the triangle is a right triangle. In this Activity, you will prove the converse of the Pythagorean Theorem by using a two-column proof. Given: . Prove: is a right triangle. Complete the proof with the correct reasons justifying each statement. Statements Reasons ______________________ ______________________ 2 2 2 a b c such thatABC 2 2 2 a b c ABC j. is a right triangle.ABC So, if a triangle has side lengths, a, b, and c units such that , then the triangle is a right triangle. 2 2 2 a b c
- 18. Course 3, Inquiry Lab after Lesson 5-5 HOW can you prove the Pythagorean Theorem and its converse? Geometry