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Review for Mastery 8-1 Similarity in Right Triangles Name Date Class LESSON Review for Mastery 8-1 Similarity in Right Triangles Altitudes and Similar Triangles The altitude to the hypotenuse of a right triangle forms two B triangles that are similar to each other and to the original triangle. B original D A C triangle D D ACABB C Similarity statement: ᭝ABC ϳ ᭝ADB ϳ ᭝BDC The geometric mean of two positive numbers is the positive square root of their product. Find the geometric mean of 5 and 20. x is the geometric Let x be the geometric mean. a x mean of a and b. __ ϭ __ 2 x b x ϭ (5)(20) Definition of geometric mean 2 2 x ϭ ab x ϭ 100 Simplify. ᎏ x ϭ 10 Find the positive square root. x ϭ ͙ab So 10 is the geometric mean of 5 and 20. Write a similarity statement comparing the three triangles in each diagram. 1. M 2. G P FH LN J Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 3. 3 and 27 4. 9 and 16 5. 4 and 5 6. 8 and 12 Copyright © by Holt, Rinehart and Winston. All rights reserved. 6 Holt Geometry Name Date Class LESSON Review for Mastery 8-1 Similarity in Right Triangles continued You can use geometric means to find side lengths in right triangles. Geometric Means Words Symbols Examples The length of the altitude to the hypotenuse of a right triangle h 6 is the geometric mean of the lengths of the two segments of x y x 9 the hypotenuse. h 2 ϭ xy h 2 ϭ xy 6 2 ϭ x(9) 36 ϭ 9x 4 ϭ x The length of a leg of a right b triangle is the geometric mean of a a the length of the hypotenuse and xy 49 the segment of the hypotenuse c 13 adjacent to that leg. a 2 ϭ xc b 2 ϭ yc a 2 ϭ xc a 2 ϭ 4(13) a 2 ϭ 52 ᎏ ᎏ a ϭ ͙52 ϭ 2 ͙13 Find x, y, and z. 7. 8. 3 y z 4 6 z x x 2 y 9. 10. z z y y 8͙ළ2 6 x 4 x 9 Copyright © by Holt, Rinehart and Winston. All rights reserved. 7 Holt Geometry LESSON Practice A LESSON Practice B 8-1 Similarity in Right Triangles 8-1 Similarity in Right Triangles In Exercises 1 and 2, fill in the blanks to complete each theorem or definition. Write a similarity statement comparing the three triangles in each diagram. 1. The geometric mean of two positive numbers is the positive square root 1. - 2. $ 3. 8 * + : of their product. altitude % & 2. The to the hypotenuse of a right triangle forms , ' 79 two triangles that are similar to each other and to the original triangle. Possible answers: 3. Write a similarity statement comparing the three triangles. " ᭝JKL ϳ ᭝JLM ϳ ᭝DEF ϳ ᭝GED ϳ ᭝WXY ϳ ᭝ZXW ϳ Be sure to write the vertices in the correct order. $ ᭝LKM ᭝GDF ᭝ZWY ᭝ABC ϳ ᭝DBA ϳ ᭝DAC !# Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. Find the geometric mean of each pair of numbers. ᎏ 4. __1 and 4 1 5. 3 and 75 15 6. 4 and 18 6͙2 4. 3 and 12 6 5. 9 and 16 12 6. 4 and 25 10 4 ᎏ _____3͙2 ᎏ Use the figure for Exercises 7–11. The big right triangle is divided by 7. __1 and 9 8. 10 and 14 2͙35 9. 4 and 12.25 7 2 2 an altitude into two smaller right triangles. The smaller triangles are also shown separated from the big triangle. All three triangles are Find x, y, and z. similar. For Exercises 7–9 complete each similarity ratio comparing 10. 11. 12. the indicated side lengths. X Y Z 10 Y Z X qi6 Z Y + X + X - - 20 C - 57 3 X A Y Y A ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ ᎏ H H ͙35 ; 2 ͙15 ; 2 ͙21 30; 10 ͙3; 20 ͙3 2; ͙15 ; ͙10 * , * , * B B 13. 15 6 14. 25 15. X 1 2 3 Z X X 65 Y 27 18 __h ϭ ___y Y Z 7. legs of triangles 2 and 3 : x h Z Y c ᎏ ᎏ ᎏ ᎏ ᎏ 8. shorter legs and the hypotenuses of triangles 1 and 3 : __a ϭ ___ 3͙10 ; 3 ͙35 ; 3 ͙14 144; 60; 156 12; 9 ͙13 ; 6 ͙13 x a 16. The Coast Guard has sent a rescue helicopter to retrieve ___b ϭ ___c 9. longer legs and the hypotenuses of the triangles 1 and 2 : y passengers off a disabled ship. The ship has called in its b position as 1.7 miles from shore. When the helicopter 10. Find the cross product of your answer to Exercise 7 to prove Theorem 8-1-2. passes over a buoy that is known to be 1.3 miles from 2 shore, the angle formed by the shore, the helicopter, h ϭ xy and the disabled ship is 90Њ. Determine what 11. Find the cross products of your answers to Exercises 8 and 9 to prove Theorem 8-1-3. the altimeter would read to the nearest foot 1.3 mi when the helicopter is directly above the buoy. a 2 ϭ xc b 2 ϭ yc 3,807 feet 1.7 mi 12. Find h, a, and b. Write your answer in simplest radical A form, if necessary. B ᎏ ᎏ H Use the diagram to complete each equation. BC h ϭ 4 a ϭ 4͙5 b ϭ 2͙5 8 2 e c d e d a E 17. __ ϭ ___ 18. _____ ϭ ___ 19. ___ϭ __ A D b e b ϩ c a c e Copyright © by Holt, Rinehart and Winston. Copyright © by Holt, Rinehart and Winston. All rights reserved. 3 Holt Geometry All rights reserved. 4 Holt Geometry LESSON Practice C LESSON Review for Mastery 8-1 Similarity in Right Triangles 8-1 Similarity in Right Triangles Find x and y. Altitudes and Similar Triangles X 1. 2. 3. Y 16 40 The altitude to the hypotenuse of a right triangle forms two B X 7 20 Y triangles that are similar to each other and to the original triangle. X Y 24 38 B original ᎏ ᎏ ᎏ D 49 576 triangle A C ___ ; ____ 19 ͙ 105 ; 19 ϩ ͙105 30; 10 ͙3 D 25 25 D 4. The arithmetic mean is also known as the average. Name the conditions under ACABB C which two nonzero, positive numbers, a and b, have equal geometric and Similarity statement: ᭝ABC ϳ ᭝ADB ϳ ᭝BDC arithmetic means. a and b have the same geometric and arithmetic means if a ϭ b. The geometric mean of two positive numbers is the positive square root of their product. 5. Sketch a right triangle in which the segments of the hypotenuse formed by the Find the geometric mean of 5 and 20. x is the geometric altitude to the hypotenuse have the same geometric and arithmetic means. Let x be the geometric mean. a x mean of a and b. __ ϭ __ 2 x b x ϭ (5)(20) Definition of geometric mean 2 2 x ϭ ab x ϭ 100 Simplify. ᎏ x ϭ 10 Find the positive square root. x ϭ ͙ab 6. Give all three angle measures of the triangle you drew in Exercise 5. 45Њ, 45Њ, 90Њ So 10 is the geometric mean of 5 and 20. 7. Name the conditions under which two nonzero, positive numbers, a and b, have an arithmetic mean that is less than their geometric mean. There are no conditions under which the arithmetic mean will be less than Write a similarity statement comparing the three triangles in each diagram. the geometric mean. 1. M 2. G P Greg is interested in buying a plot of land. He is looking at a plot in the shape of a right triangle. A dirt road makes an altitude to the longest side of the plot and cuts FH J the longest side into two parts that measure 65 feet and 83 feet. LN 2 8. Find the area of the land to the nearest square foot. 5435 ft ᭝MNL ϳ ᭝NPL ϳ ᭝MPN ᭝FGH ϳ ᭝FJG ϳ ᭝GJH 9. Find the perimeter of the land to the nearest foot. 357 ft Find the geometric mean of each pair of numbers. If necessary, give the answer " Use the figure for Exercises 11–12. in simplest radical form. IN The figure shows ᭝ABC. 3. 3 and 27 4. 9 and 16 AB ϭ 3 in. and AC ϭ 4 in. !#IN ‹__› 10. Point D is placed so that ЄCBD is a right angle and D is on AC . 9 12 ___15 ϭ 3 __3 in. Find BD. 4 4 ‹__› 5. 4 and 5 6. 8 and 12 11. Point E is placed so that ЄBCE is a right angle and E is on AB. ___20 __2 Find CE. ϭ 6 in. ᎏ ᎏ 3 3 2 ͙5 4 ͙6 ᎏ ___5 ͙193 in. Ϸ 5 __3 in. 12. Find DE. 12 4 Copyright © by Holt, Rinehart and Winston. Copyright © by Holt, Rinehart and Winston. All rights reserved. 5 Holt Geometry All rights reserved. 6 Holt Geometry Copyright © by Holt, Rinehart and Winston. 001-062_Go08an_CRF_c08.indd 6 4/13/07 10:56:04 AM All rights reserved.
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