Practice 4-4 (DNG page 327) [PDF]

Similarity in Right Triangles

1/15/2014 5:52 PM

PRACTICE 7-4 (DNG PAGE 385) 7-4 Guided Problem Solving (page 386)

Geometry/CBautista

Practice 7-4: SIMILARITY IN RIGHT TRIANGLES

Find the geometric mean of each pair of numbers.

1. 32 and 8

2. 4 and 16

GM  16 4. 2 and 22

 

GM  2 22

  

GM  2 2 11

GM  2 11

 



GM  4 16

GM  11 7

GM  8

GM  77

5. 10 and 20

6. 6 and 30

  10102 

 

GM  10 20

GM  6 30

GM 

GM  6 6 5

GM  10 2

  

GM  6 5

Geometry/CBautista



GM  32 8

3. 11 and 7

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Algebra

DNG - page 385

DNG - page 385

Practice 7-4: SIMILARITY IN RIGHT TRIANGLES

Refer to the figure to complete each proportion.

Algebra

b

¬

h h

x

7.

10.

𝒙 𝒉 𝒂 𝒄

= =

?h 𝒚 𝒚 ?a

8.

11.

𝒂 𝒃 𝒂 𝒄

= =

?y 𝒉 𝒉 ?b

a

¬y 9.

12.

(original figure)

𝒂 𝒃 𝒃 𝒙

= =

𝒉 ?x ?c 𝒃

Practice 7-4: SIMILARITY IN RIGHT TRIANGLES

Algebra

DNG - page 385

Find the values of the variables. 1/14/2014 7:47 PM

Solve for y:

Solve for y:

Solve for x:

x  312

y  912

y  412

x  1216

x  334 

y  3 43

y  44 3

x  32 

y  3(2) 3

x6

y6 3

y4 3

x  4 43

x  42 3

x 8 3

Geometry/CBautista

Solve for x:

Practice 7-4: SIMILARITY IN RIGHT TRIANGLES Algebra

Find the values of the variables.

Solve for y: Solve for x:

x 3



3

y  511 y  55

2 Solve for x:

x

9 2

DNG page 385

x  516

x4 5

Solve for z:

Solve for y:

z  12

y  23

z 2 Solve for x:

x  31 x 3

y 6

Practice 7-4: SIMILARITY IN RIGHT TRIANGLES Find the values of the variables.

Algebra

Solve for x:

Solve for y: (by Pythagorean Theorem)

y 1  3 2

DNG - page 385

2

2

y  3 1 2

y 8 y2 2

2

y  1 x 

8 x x 8

Solve for z:

z  x x  1

z  88  1 z  429

z  23 2 z 6 2

Practice 7-4: SIMILARITY IN RIGHT TRIANGLES 19.The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments 6 in. and 10 in. long. Find the length h of the altitude.

Use Corollary 1 of Theorem 7-3.

h  610

h  2325

h 6

10

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h  2 15 in

Geometry/CBautista

7-4 • Guided Problem Solving DNG page 386 For a right triangle, denote lengths as follows: 𝒍𝟏 and 𝒍𝟐 the legs, h the hypotenuse AB, a the altitude, and 𝒉𝟏 and 𝒉𝟐 the hypotenuse segments determined by the altitude. For h = 2 and 𝒉𝟏 = 1, find the other four measures. Use simplest radical form.

1=

=1

Read and Understand h=2 1. Using the letters A, B, C, and D, label the figure as illustrated. ABC, ACD, BCD What are the similar triangles you will use to solve this problem? _______________

1 2. If h = 2 and h1 = 1, you can solve for h2. What is h2? h2 = ______ Use ACD and BCD to solve for a as follows.

=

𝒂 ( 1)

1 𝒂𝟐 = _______ 1 𝒂 = ________

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3. Write a proportion.

a A

¬

(1 ) 𝒂

h1=1

4. Use the Cross-Product Property. 5. Take the square root.

Geometry/CBautista

C

C

D

a

¬

D

h2=1

B

7-4 • Guided Problem Solving DNG page 386 For a right triangle, denote lengths as follows: 𝒍𝟏 and 𝒍𝟐 the legs, h the hypotenuse AB, a the altitude, and 𝒉𝟏 and 𝒉𝟐 the hypotenuse segments determined by the altitude. For h = 2 and 𝒉𝟏 = 1, find the other four measures. Use simplest radical form.

Read and Understand… h=2

To solve for 𝒍𝟏 , compare ABC and ACD. ( 2 )

=

( 1) 𝒍𝟏

𝒍𝟏

6. Write a proportion.

2 𝒍𝟏 𝟐 = ________

7. Use the Cross-Product Property.

𝟐 𝒍𝟏 = ________

8. Take the square root.

A

𝟐 9. Finally, 𝒍𝟐 is found in a similar manner. What is 𝒍𝟐 ? 𝒍𝟐 = _______ Look Back and Check 10. Use a ruler to construct triangles with the dimensions found. Are your answers correct? YES. Solve Another Problem 11. If, instead of h and h1, values for 𝒍𝟏 and 𝒍𝟐 were given originally, could you solve for the other parts with that information? NO. (This would require the Pythagorean Theorem.) 1/15/2014 5:47 PM

Geometry/CBautista

a

¬

𝒍𝟏

C

h1=1

D