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Geometric Mean and Pythagorean Theorem Lesson

This lesson focuses on finding the geometric mean, using the Pythagorean theorem, and verifying right triangles. Students will practice solving problems involving right triangles and applying the Pythagorean theorem. Includes proofs and examples.

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Geometric Mean and Pythagorean Theorem Lesson

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  1. D.N.A. 1) Find the geometric mean of 8 and 12. 2) The geometric mean of 8 and x is 11. Find x. Simplify each expression.

  2. 8 – 2: The Pythagorean Theorem Textbook pp. 440 - 446

  3. Use the Pythagorean Theorem and its converse. • Pythagorean triple Standard 12.0 Students find and use measures of sides and of interior and exterior angles of trianglesand polygons to classify figures and solve problems.(Key) Standard 14.0 Students prove the Pythagorean theorem. (Key) Standard 15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. Lesson 2 MI/Vocab

  4. Pythagorean Theorem • In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. a2 + b2 = c2 Click here for the Pythagorean Proof

  5. Find x. Round your answer to the nearest tenth. • A • B • C • D A. 17 B. 12.7 C. 11.5 D. 13.2 Lesson 2 CYP2

  6. Verify a Triangle is a Right Triangle COORDINATE GEOMETRY Verify that ΔABC is a right triangle. Use distance formula on all 3 sides then the Pythagorean theorem. Lesson 2 Ex3

  7. 10 2 6 4 6 4 Verify a Triangle is a Right Triangle COORDINATE GEOMETRY Verify that ΔABC is a right triangle. Lesson 2 Ex3

  8. COORDINATE GEOMETRY Is ΔRST a right triangle? • A • B • C • A. yes • B. no • cannot be determined Lesson 2 CYP3

  9. Pythagorean Triples A. Determine whether 9, 12, and 15are the sides of a right triangle. Then state whether they form a Pythagorean triple. Since the measure of the longest side is 15, 15 must be c. Let a and b be 9 and 12. Pythagorean Theorem Simplify. Add. Lesson 2 Ex4

  10. Homework Chapter 8-2 • Pg 444: #1 – 3, 6 – 26

  11. The Pythagorean Theorem: (Area of green square) + (Area of red square) = Area of the blue square height • We start with half the red square, which has • Area = ½ base x height base • We move one vertex while maintaining the base & height, so that the area remains the same. This is called a SHEAR. base • We rotate this triangle, which does not change its area. height • We mark the base and height for this triangle.

  12. The Pythagorean Theorem: (Area of green square) + (Area of red square) = Area of the blue square height • We start with half the red square, which has • Area = ½ base x height Half the red square. base • We move one vertex while maintaining the base & height, so that the area remains the same. This is called a SHEAR. • We rotate this triangle, which does not change its area. • We mark the base and height for this triangle. • We now do a shear on this triangle, keeping the same area. Remember that this pink triangle is half the red square.

  13. The Pythagorean Theorem: (Area of green square) + (Area of red square) = Area of the blue square • The other half of the red square has the same area as this pink triangle, so if we copy and rotate it, we get this. So, together these two pink triangles have the same area as the red square. Half the red square. Shear • We now take half of the green square and transform it the same way. Half the red square. Half the green square. Half the green square. We end up with this triangle, which is half of the green square. Rotate • The other half of the green square would give us this. Shear • Together, they have they same • area as the green square. So, we have shown that the red & green squares together have the same area as the blue square.

  14. The Pythagorean Theorem: We’ve Proven the Pythagorean Theorem (click to return) (Area of green square) + (Area of red square) = Area of the blue square • The other half of the red square has the same area as this pink triangle, so if we copy and rotate it, we get this. So, together these two pink triangles have the same area as the red square. Half the red square. • We now take half of the green square and transform it the same way. Shear Half the red square. Half the green square. We end up with this triangle, which is half of the green square. Half the green square. • The other half of the green square would give us this. Rotate • Together, they have they same • area as the green square. Shear So, we have shown that the red & green squares together have the same area as the blue square. We’ve PROVEN the Pythagorean Theorem!

  15. Find the missing side of the triangle. 1) 2) 3)

  16. Hypotenuse and Segment of Hypotenuse Find c and d in ΔJKL. = 20 Lesson 1 Ex4

  17. 25 Hypotenuse and Segment of Hypotenuse Find c and d in ΔJKL. = 20 = 11.2 Lesson 1 Ex4

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