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Proving Triangles Similar

ABX ~ CDX by SAS ~ Theorem. MRT ~ XRW by AA ~ Postulate. Proving Triangles Similar. Lesson 7-3. Lesson Quiz. Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain. 1. 2. 3. 4. Find the value of x.

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Proving Triangles Similar

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  1. ABX ~ CDX by SAS ~ Theorem MRT ~ XRW by AA ~ Postulate Proving Triangles Similar Lesson 7-3 Lesson Quiz Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain. 1.2. 3. 4. Find the value of x. 5.When a 6 ft tall man casts a shadow 18 ft long, a nearby tree casts a shadow 93 ft long. How tall is the tree? The congruent angle is not included between the proportional sides, so you cannot conclude that the triangles are similar. 16 31 ft 7-4

  2. Similarity in Right Triangles Lesson 7-4 Check Skills You’ll Need (For help, go to Lesson 7-1 and page 390.) Solve each proportion. 1. = 2. = 3. = 4. = 5. = 6. = 7. = 8.= 9. Draw a right triangle. Label the triangle ABC with right angle C. Draw the altitude to the hypotenuse. Label the altitude CD. Name the two smaller right triangles that are formed. x 18 8 24 2 x 3 7 1518 4 x 5117 x 13 – – – – 4 x 10 5 3 9 m 8 w20 2 9 9 27 6 a – – – – Check Skills You’ll Need 7-4

  3. x 8 18 24 18 24 8 1 18 3 1. Multiply both sides of = by 8; x =  = = 6 2. Multiply both sides of = by 7; x =  = or 4 3. = ; (15)x = 18(4) (Cross Products);Simplify: 15x = 72; divide by 15: x = = or 4 4. = ; (51)13 = 17(x) (Cross Products);Simplify: 663 = 17x; divide by 17: x = 39 5. Multiply both sides of = by 5; x =  = = 2 2 3 2 3 x 7 2 3 7 1 14 3 15 4 18 x 4 5 72 15 24 5 51 x 17 13 x 5 4 10 5 1 4 2 4 10 Similarity in Right Triangles Lesson 7-4 Check Skills You’ll Need Solutions 7-4

  4. 6. = ; (3)8 = 9(m) (Cross Products);Simplify: 24 = 9m; divide by 9: m = = or 2 7. Multiply both sides of = by 2; w =  = or 4 8. Simplify in = ; = ; 3(a) = 27(2) (Cross Products);Simplify: 3a = 54; divide by 3: a = 18 9. The altitude divides ABC into two smallerright triangles, ADC and CDB. Solutions (continued) 9 8 24 9 8 3 2 3 w 2 20 9 20 9 2 1 40 9 4 9 9 6 9 6 27 a 3 2 27 a 3 m Similarity in Right Triangles Lesson 7-4 Check Skills You’ll Need 7-4

  5. Similarity in Right Triangles Lesson 7-4 Notes 7-4

  6. Similarity in Right Triangles Lesson 7-4 Notes The geometric mean of two positive numbers a and b is the positive number x such that Note that 7-4

  7. Similarity in Right Triangles Lesson 7-4 Notes 7-4

  8. Similarity in Right Triangles Lesson 7-4 Notes 7-4

  9. 3 x x 12 = Write a proportion. x2 = 36 Cross-Product Property x = 36 Find the positive square root. Similarity in Right Triangles Lesson 7-4 Additional Examples Finding the Geometric Mean Find the geometric mean of 3 and 12. x = 6 The geometric mean of 3 and 12 is 6. Quick Check 7-4

  10. Use Corollary 1 of Theorem 7-3 (The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse) to find x. 2 6 6 x = Write a proportion. 2x = 36 Cross-Product Property x = 18 Similarity in Right Triangles Lesson 7-4 Additional Examples Applying Corollaries Solve for x and y. 7-4

  11. x y y 2 + x = Write a proportion. y 2 + 18 18 y = Substitute 18 for x. y2 = 360 Cross-Product Property. y = 360 Find the positive square root. y = 6 10 Write in the simplest radical form. Similarity in Right Triangles Lesson 7-4 Additional Examples (continued) Use Corollary 2 of Theorem 7-3 (The altitude to the hypotenuse of a right triangle intersects it so that the length of each leg is the geometric mean of the length of its adjacent segment of the hypotenuse and the length of the entire hypotenuse) to find y. Quick Check 7-4

  12. x + 192 240 240 192 = Write a proportion. 192(x +192) = 2402 Use the Cross-Product Property. 192x + 36,864 = 57,600 Use the Distributive Property. 192x = 20,736 Solve for x. x = 108 Similarity in Right Triangles Lesson 7-4 Additional Examples Real-World Connection At a golf course, Maria drove her ball 192 yd straight toward the cup. Her brother Gabriel drove his ball straight 240 yd, but not toward the cup. The diagram shows the results. Find x and y, their remaining distances from the cup. Use Corollary 2 of Theorem 7-3 to solve for x. 7-4

  13. x + 192 y y x = Write a proportion. 108 + 192 y = Substitute. 300 y = Simplify. y2 = 32,400 Use the Cross-Product Property. y= 32,400 Find the positive square root. y= 180 y 108 y 108 Similarity in Right Triangles Lesson 7-4 Additional Examples (continued) Use Corollary 2 of Theorem 7-3 to solve for y. Maria’s ball is 108 yd from the cup, and Gabriel’s ball is180 yd from the cup. Quick Check 7-4

  14. 2 30 5 3 width = 28 2 ft, height = 14 2 ft Similarity in Right Triangles Lesson 7-4 Lesson Quiz 1. Find the geometric mean of 32 and 2. 2. Find the geometric mean of 6 and 20. 3. Solve for x. 4. Solve for x and y. 5. The roof of a house forms a right angle, with each side of the roof measuring 28 ft in length. Find the width and the height of the roof. 8 x = 9, y = 16 7-4

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