7.3 Similar Right Triangles

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Investigating g Geometr IVIY Use before Lesson 7.

E X L O E E 1 ompare right triangles E 2 1 7 6 2 raw a diagonal raw a diagonal on our rectangular piece of paper to form two congruent right triangles.

Label the angles and color the triangles as shown. rrange the triangles rrange the triangles so 1,, and 7 are on top of each other as shown.

1 Investigating g Geometr IVIY Use before Lesson imilar ight riangles MEIL rectangular piece of paper ruler scissors colored pencils Q U E I O N How are geometric means related to the altitude of a right triangle? E X L O E E 1 ompare right triangles E raw a diagonal raw a diagonal on our rectangular piece of paper to form two congruent right triangles. E raw an altitude Fold the paper to make an altitude to the hpotenuse of one of the triangles. E ut and label triangles ut the rectangle into the three right triangles that ou drew. Label the angles and color the triangles as shown. rrange the triangles rrange the triangles so 1,, and 7 are on top of each other as shown. W O N L U I O N Use our observations to complete these eercises 1. How are the two smaller right triangles related to the large triangle? 2. Eplain how ou would show that the green triangle is similar to the red triangle.. Eplain how ou would show that the red triangle is similar to the blue triangle.. he geometric mean of a and b is if a b. Write a proportion involving the side lengths of two of our triangles so that one side length is the geometric mean of the other two lengths in the proportion. hapter 7 ight riangles and rigonometr

7. Use imilar ight riangles efore You identified the altitudes of a triangle. Now You will use properties of the altitude of a right triangle. Wh?

72 When the altitude is drawn to the hpotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other.

If the altitude is drawn to the hpotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

Each triangle has a right angle and each triangle includes. he triangles are similar b the imilarit ostulate. Use similar reasoning to show that n, n.

2 7. Use imilar ight riangles efore You identified the altitudes of a triangle. Now You will use properties of the altitude of a right triangle. Wh? o ou can determine the height of a wall, as in Eample. Ke Vocabular altitude of a triangle, p. 20 geometric mean, p. similar polgons, p. 72 When the altitude is drawn to the hpotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other. HEOEM HEOEM 7. If the altitude is drawn to the hpotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. n, n, n, n, and n, n. For Your Notebook roof: below; E., p. 6 lan for roof of heorem 7. First prove that n, n. Each triangle has a right angle and each triangle includes. he triangles are similar b the imilarit ostulate. Use similar reasoning to show that n, n. o show, n, begin b showing > because the are both complementar to. Each triangle also has a right angle, so ou can use the imilarit ostulate. E X M L E 1 Identif similar triangles Identif the similar triangles in the diagram. U olution ketch the three similar right triangles so that the corresponding angles and sides have the same orientation. U U c n U, n U, n 7. Use imilar ight riangles

E X M L E 2 Find the length of the altitude to the hpotenuse WIMMING OOL he diagram below shows a cross-section of a swimming pool. What is the maimum depth of the pool? 12 in. 16 in. h M in. 6 in.

h in. M n, nm, nm VOI EO Notice that if ou tried to write a proportion using nm and nm, there would be two unknowns, so ou would not be able to solve for h. E 2 Find the value of h.

3 E X M L E 2 Find the length of the altitude to the hpotenuse WIMMING OOL he diagram below shows a cross-section of a swimming pool. What is the maimum depth of the pool? 12 in. 16 in. h M in. 6 in. olution E 1 Identif the similar triangles and sketch them. 16 in. 12 in. 6 in. 12 in. 6 in. M h in. h in. M n, nm, nm VOI EO Notice that if ou tried to write a proportion using nm and nm, there would be two unknowns, so ou would not be able to solve for h. E 2 Find the value of h. Use the fact that n, nm to write a proportion. M orresponding side lengths of similar triangles are in proportion. h 12 ubstitute h 6(12) ross roducts ropert E h ø olve for h. ead the diagram above. You can see that the maimum depth of the pool is h 1, which is about inches. c he maimum depth of the pool is about 107 inches. at classzone.com GUIE IE for Eamples 1 and 2 Identif the similar triangles. hen find the value of. 1. E G H F 2. J 1 K L M 0 hapter 7 ight riangles and rigonometr

4 GEOMEI MEN In Lesson 6.1, ou learned that the geometric mean of two numbers a and b is the positive E YMOL emember that an altitude is defined as a segment. o, refers to an altitude in n and refers to its length. number such that a b. onsider right n. From heorem 7., ou know that altitude forms two smaller triangles so that n, n, n. I II III Notice that is the longer leg of n and the shorter leg of n. When ou write a proportion comparing the leg lengths of n and n, ou can see that is the geometric mean of and. s ou see below, and are also geometric means of segment lengths in the diagram. roportions Involving Geometric Means in ight n length of shorter leg of I length of shorter leg of II length of hpotenuse of III length of hpotenuse of I length of hpotenuse of III length of hpotenuse of II length of longer leg of I length of longer leg of II length of shorter leg of III length of shorter leg of I length of longer leg of III length of longer leg of II E X M L E Use a geometric mean Find the value of. Write our answer in simplest radical form. olution EVIEW IMILIY Notice that nq and nq both contain the side with length, so these are the similar pair of triangles to use to solve for. E 1 raw the three similar triangles. E 2 Write a proportion. length of hp. of nq length of shorter leg of nq length of hp. of n Q length of shorter leg of nq ubstitute ross roducts ropert Ï 27 Ï ake the positive square root of each side. implif. 7. Use imilar ight riangles 1

HEOEM For Your Notebook HEOEM 7.6 Geometric Mean (ltitude) heorem WIE OOF In Eercise 2 on page, ou will use the geometric mean theorems to prove the thagorean heorem.

roof: E. 6, p. 6 HEOEM 7.7 Geometric Mean (Leg) heorem In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments.

5 HEOEM For Your Notebook HEOEM 7.6 Geometric Mean (ltitude) heorem WIE OOF In Eercise 2 on page, ou will use the geometric mean theorems to prove the thagorean heorem. In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. he length of the altitude is the geometric mean of the lengths of the two segments. roof: E. 6, p. 6 HEOEM 7.7 Geometric Mean (Leg) heorem In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. he length of each leg of the right triangle is the geometric mean of the lengths of the hpotenuse and the segment of the hpotenuse that is adjacent to the leg. roof: E. 7, p. 6 and E X M L E Find a height using indirect measurement OK LIMING WLL o find the cost of installing a rock wall in our school gmnasium, ou need to find the height of the gm wall. You use a cardboard square to line up the top and bottom of the gm wall. Your friend measures the vertical distance from the ground to our ee and the distance from ou to the gm wall. pproimate the height of the gm wall.. ft w ft ft olution heorem 7.6, ou know that. is the geometric mean of w and. w. Write a proportion.. w ø 1. olve for w. c o, the height of the wall is 1 w ø feet. GUIE IE for Eamples and. In Eample, which theorem did ou use to solve for? Eplain.. Mar is. feet tall. How far from the wall in Eample would she have to stand in order to measure its height? 2 hapter 7 ight riangles and rigonometr

7. EXEIE KILL IE HOMEWOK KEY WOKE-OU OLUION on p. W1 for Es., 1, and 2 NIZE E IE Es. 2, 1, 20, 1, and 1. VOULY op and complete: wo triangles are?

EXMLE 1 on p. for Es. IENIFYING IMIL INGLE Identif the three similar right triangles in the given diagram.. F E. M H G L N K EXMLE 2 on p. 0 for Es.

2 ft 10 ft 76 ft 2 ft. ft EXMLE and on pp. 1 2 for Es. 1 OMLEING OOION Write a similarit statement for the three similar triangles in the diagram.

6 7. EXEIE KILL IE HOMEWOK KEY WOKE-OU OLUION on p. W1 for Es., 1, and 2 NIZE E IE Es. 2, 1, 20, 1, and 1. VOULY op and complete: wo triangles are? if their corresponding angles are congruent and their corresponding side lengths are proportional. 2. WIING In our own words, eplain geometric mean. EXMLE 1 on p. for Es. IENIFYING IMIL INGLE Identif the three similar right triangles in the given diagram.. F E. M H G L N K EXMLE 2 on p. 0 for Es. 7 FINING LIUE Find the length of the altitude to the hpotenuse. ound decimal answers to the nearest tenth ft 76 ft 26.6 ft. ft 1.2 ft 10 ft 76 ft 2 ft. ft EXMLE and on pp. 1 2 for Es. 1 OMLEING OOION Write a similarit statement for the three similar triangles in the diagram. hen complete the proportion.. XW? ZW YW.? Q Q Q 10. EF EG EG? W Y E H F X Z G EO NLYI escribe and correct the error in writing a proportion for the given diagram w w z z w 1 v v z g h e d d f d e f 7. Use imilar ight riangles

7 FINING LENGH Find the value of the variable. ound decimal answers to the nearest tenth z MULILE HOIE Use the diagram at the right. ecide which proportion is false. 20. MULILE HOIE In the diagram in Eercise 1 above, 6 and 1. Find. If necessar, round to the nearest tenth LGE Find the value(s) of the variable(s) b 1 2. a z UING HEOEM ell whether the triangle is a right triangle. If so, find the length of the altitude to the hpotenuse. ound decimal answers to the nearest tenth FINING LENGH Use the Geometric Mean heorems to find and HLLENGE raw a right isosceles triangle and label the two leg lengths. hen draw the altitude to the hpotenuse and label its length. Now draw the three similar triangles and label an side length that is equal to either or. What can ou conclude about the relationship between the two smaller triangles? Eplain. WOKE-OU OLUION on p. W1 NIZE E IE

OLEM OLVING 2. OGHOUE he peak of the doghouse shown forms a right angle. Use the given dimensions to find the height of the roof. 1. ft 1. ft EXMLE on p. 2 for Es. 0 1 0.

Mar measures the vertical distance from the ground to our ee and the distance from ou to the monument. pproimate the height of the monument (as shown at the left below). 1.

He etends the rope to the cardboard square he is holding lined up to the top and bottom of the monument. Use the information in the diagram above to approimate the height of the monument.

8 OLEM OLVING 2. OGHOUE he peak of the doghouse shown forms a right angle. Use the given dimensions to find the height of the roof. 1. ft 1. ft EXMLE on p. 2 for Es MONUMEN You want to determine the height of a monument at a local park. You use a cardboard square to line up the top and bottom of the monument. Mar measures the vertical distance from the ground to our ee and the distance from ou to the monument. pproimate the height of the monument (as shown at the left below). 1. HO EONE aul is standing on the other side of the monument in Eercise 0 (as shown at the right above). He has a piece of rope staked at the base of the monument. He etends the rope to the cardboard square he is holding lined up to the top and bottom of the monument. Use the information in the diagram above to approimate the height of the monument. o ou get the same answer as in Eercise 0? Eplain. 2. OVING HEOEM 7.1 Use the diagram of n. op and complete the proof of the thagorean heorem. b f c e GIVEN c In n, is a right angle. OVE c c 2 a 2 1 b 2 a EMEN 1. raw n. is a right angle. 2. raw a perpendicular from to.. c a a e and c b b f. ce a 2 and cf b 2. ce 1 b 2? 1 b 2 6. ce 1 cf a 2 1 b 2 7. c(e 1 f ) a 2 1 b 2. e 1 f?. cp c a 2 1 b c 2 a 2 1 b 2 EON 1.? 2. erpendicular ostulate.?.?. ddition ropert of Equalit 6.? 7.?. egment ddition ostulate.? 10. implif. 7. Use imilar ight riangles

. MULI-E OLEM Use the diagram. a. Name all the altitudes in negf. Eplain. b. Find FH. c. Find the area of the triangle. F E H 7 G. EXENE EONE Use the diagram. a. ketch the three similar triangles in the diagram.

9 . MULI-E OLEM Use the diagram. a. Name all the altitudes in negf. Eplain. b. Find FH. c. Find the area of the triangle. F E H 7 G. EXENE EONE Use the diagram. a. ketch the three similar triangles in the diagram. Label the vertices. Eplain how ou know which vertices correspond. b. Write similarit statements for the three triangles. c. Which segment s length is the geometric mean of and Q? Eplain our reasoning. OVING HEOEM In Eercises 7, use the diagram and GIVEN statements below. GIVEN c n is a right triangle. ltitude is drawn to hpotenuse.. rove heorem 7. b using the lan for roof on page. 6. rove heorem 7.6 b showing. 7. rove heorem 7.7 b showing and.. HLLENGE he harmonic mean of a and b is 2ab. he Greek a 1 b mathematician thagoras found that three equall taut strings on stringed instruments will sound harmonious if the length of the middle string is equal to the harmonic mean of the lengths of the shortest and longest string. a. Find the harmonic mean of 10 and 1. b. Find the harmonic mean of 6 and 1. c. Will equall taut strings whose lengths have the ratio : 6 : sound harmonious? Eplain our reasoning. MIXE EVIEW EVIEW repare for Lesson 7. in Es. 6. implif the epression. (p. 7). Ï 27p Ï 2 0. Ï p Ï Ï p Ï 7 2. Ï 1p Ï. Ï 7. Ï Ï Ï 2 ell whether the lines through the given points are parallel, perpendicular, or neither. Justif our answer. (p. 171) 7. Line 1: (2, ), (, 2). Line 1: (0, 2), (21, 21) : Line 1: (1, 7), (, 7) Line 2: (, ), (21, 1) Line 2: (, 1), (1, 2) Line 2: (, 2), (7, ) 6 EX IE for Lesson 7., p. 0 ONLINE QUIZ at classzone.com

9.4. Perform Rotations. Draw a rotation. STEP 1 Draw a segment from A to P. STEP 2 Draw a ray to form a 1208 angle with } PA.

9.4. Perform Rotations. Draw a rotation. STEP 1 Draw a segment from A to P. STEP 2 Draw a ray to form a 1208 angle with } PA. 40 40 50 30 9.4 erform otations efore You rotated figures about the origin. Now You will rotate figures about a point. Wh? So ou can classif transformations, as in Es. 3 5. Ke Vocabular center of rotation