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Splash Screen. Five-Minute Check (over Chapter 7) CCSS Then/Now New Vocabulary Key Concept: Geometric Mean Example 1: Geometric Mean Theorem 8.1 Example 2: Identify Similar Right Triangles Theorems: Right Triangle Geometric Mean Theorems

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 7) CCSS Then/Now New Vocabulary Key Concept: Geometric Mean Example 1: Geometric Mean Theorem 8.1 Example 2: Identify Similar Right Triangles Theorems: Right Triangle Geometric Mean Theorems Example 3: Use Geometric Mean with Right Triangles Example 4: Real-World Example: Indirect Measurement Lesson Menu

  3. Solve the proportion A.15 B.16.5 C. D.18 5-Minute Check 1

  4. The triangles at the right are similar. Find x and y. A.x = 16.1, y = 1.6 B.x = 15.6, y = 2.1 C.x = 7.8, y = 8.4 D.x = 17.6, y = 3.7 5-Minute Check 2

  5. Determine whether the triangles are similar. If so, write a similarity statement. A. yes, ΔABC ~ ΔEDF B. yes, ΔABC ~ ΔDEF C. yes, ΔABC ~ ΔEFD D. No, sides are not proportional. 5-Minute Check 3

  6. Find the perimeter of ΔDEF if ΔABC ~ ΔDEF, AB = 6.3, DE = 15.75, and the perimeter of ΔABC is 26.5. A. 66.25 B. 64.5 C. 12.4 D. 10.6 5-Minute Check 4

  7. Figure JKLM has a perimeter of 56 units. After a dilation with a scale factor of 2, what will the perimeter of figure J'K'L'M' be? A. 28 units B. 54 units C. 58 units D. 112 units 5-Minute Check 5

  8. Content Standards G.SRT.4 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 7 Look for and make use of structure. 3 Construct viable arguments and critique the reasoning of others. CCSS

  9. You used proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles. • Find the geometric mean between two numbers. • Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. Then/Now

  10. geometric mean Vocabulary

  11. Concept

  12. Geometric Mean Find the geometric mean between 2 and 50. Let x represent the geometric mean. Definition of geometric mean Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10. Example 1

  13. A. Find the geometric mean between 3 and 12. A. 3.9 B. 6 C. 7.5 D. 4.5 Example 1

  14. Concept

  15. Identify Similar Right Triangles Write a similarity statement identifying the three similar triangles in the figure. Separate the triangles into two triangles along the altitude. Example 2

  16. Identify Similar Right Triangles Then sketch the three triangles, reorienting the smaller ones so that their corresponding angles and sides are in the same position as the original triangle. Answer: So, by Theorem 8.1, ΔEGF ~ ΔFGH ~ ΔEFH. Example 2

  17. Write a similarity statement identifying the three similar triangles in the figure. A.ΔLNM ~ ΔMLO ~ ΔNMO B.ΔNML ~ ΔLOM ~ ΔMNO C.ΔLMN ~ ΔLOM ~ ΔMON D.ΔLMN ~ ΔLMO ~ ΔMNO Example 2

  18. Concept

  19. Use Geometric Mean with Right Triangles Find c, d, and e. Example 3

  20. Use Geometric Mean with Right Triangles Since e is the measure of the altitude drawn to the hypotenuse of right ΔJKL, e is the geometric mean of the lengths of the two segments that make up the hypotenuse, JM and ML. Geometric Mean (Altitude) Theorem Substitution Simplify. Example 3

  21. Since d is the measure of leg JK, d is the geometric mean of JM, the measure of the segment adjacent to this leg, and the measure of the hypotenuse JL. Use Geometric Mean with Right Triangles Geometric Mean (Leg) Theorem Substitution Use a calculator to simplify. Example 3

  22. Since c is the measure of leg KL, c is the geometric mean of ML, the measure of the segment adjacent to KL, and the measure of the hypotenuse JL. Use Geometric Mean with Right Triangles Geometric Mean (Leg) Theorem Substitution Use a calculator to simplify. Answer:e = 12, d≈ 13.4, c ≈ 26.8 Example 3

  23. Find e to the nearest tenth. A. 13.9 B. 24 C. 17.9 D. 11.3 Example 3

  24. Indirect Measurement KITES Ms. Alspach is constructing a kite for her son. She has to arrange two support rods so that they are perpendicular. The shorter rod is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod? Example 4

  25. Let be the altitude drawn from the right angle of ΔWYZ. Indirect Measurement Draw a diagram of one of the right triangles formed. Example 4

  26. Indirect Measurement Geometric Mean (Altitude) Theorem Substitution Square each side. Divide each side by 7.25. Answer: The length of the long rod is 7.25 + 25.14, or about 32.39 inches long. Example 4

  27. AIRPLANES Ajetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, at point E.If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot? A. 68.3 ft B. 231.3 ft C. 273.1 ft D. 436.1 ft Example 4

  28. Assignment: • Pg. 541/ 1-22, 26-37, 58-74

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