8.6 Proportions and Similar Triangles

Triangle Proportionality Theorem If a line parallel to one side of a triangle

intersects the other two sides, then it divides the two sides proportionally.

If TU ll QS, then R

T

QS

U

USRU

TQRT

>

>

Given AB ll ED, find EA

8

4

12

C

D

B

E

A

EA12

48

8 EA = 4(12) 8 EA = 48 EA = 6

>

>

Converse of the Triangle Proportionality Theorem

Determining Parallels

Given the diagram, determine whether MN ||

GH.

N HL

MG

1648

21

56

LM = LN NOTICE:WE ARE COMPARINGMG NH SEGMENTS TO SEGMENTS!

56 = 48 21 16

Since the cross-products are not equal, the ratios are not the same, and the segmentMN is not parallel to segment GH.

56(16) ≠ 48(21)

Theorems If three parallel lines intersect two transversals,

then they divide the transversals proportionally.

UW = VX

WY XZ

U W Y

V XZ

Theorems If a ray bisects an angle of a triangle, then it

divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

If CD bisects <ACB,Then AD = DB

AC CB

A

D

BC

Given CD=14, find DB.

C D B

915

A

X9

1415

15 X = 9(14)

15 X = 126

X = 8.4 14 X

Solve

20

108

f

8 = 10 f 20

160 = 10f 16 = f

Solve.

32

14

10

p

10 = p24 32

24p = 320 p ≈ 13.3

Solve.

32 14

18

s

18 = 3232 s

18s = 1024 s = 56 8/9

23

12

25

x

x = 1225 23

23x = 300 x ≈ 13

END OF 8.6

8.7 Dilations

Warmup

Find the value of the variable.

8

6

5 t+3

35

86

t

6(t+3) = 5(8)6t + 18 = 40 6t = 22 t ≈ 3.67

Vocabulary

Dilation: type of transformation which contains reduction and enlargement.

Reduction: 0< scale factor <1 Enlargement: scale factor >1

preimageimagek '

Examples on Identifying Dilations

Identify the dilation and find its scale factor.

preimageimagek

C

P’

P

2

552

k

Since k<1, it’s a reductionAnd scale factor is 2:5

image

preimage

Dilation in a coordinate plane.

Draw a dilation of rectangle ABCD with A(2,2), B(6,2), C(6,4), D(2,4). Name the dilation EFGH. Use the origin as the center and use a scale factor of ½.

How does the perimeter of the preimage compare to the perimeter of the image?

The perimeter of EFGH is ½ the perimeter of ABCD.

Dilation

http://www.geogebra.org/webstart/geogebra.html

Dilation in a coordinate plane

Draw a dilation of rectangle ABCD with A(2,2), B(6,2), C(6,4), D(2,4). Use the origin as the center and use a scale factor of 2.

dilation

http://www.geogebra.org/webstart/geogebra.html

Ch 8 Review

Day 4 Part 2

Review 1

Find the geometric mean of 4 and 6

Find the geometric mean of 32 and 96

Review 2

What is the perimeter of ΔABC?

AD

C

30

B

20

8

7

10

Review 3

Find the value of JN.

L

N

J

M

K

16

3

18

Review 4

Simplify the followings.

ydft

520

dayhr15

mcm410

Review 5

Identify the dilation and find its scale factor.

C

P’

P

6

14

Review 6

The ratio of AB : BC is 3 : 8. Solve for x.

A B

CD

6

X

Review 7

The perimeter of a rectangle is 54. The ratio of the lengths of the sides is 2:7. What are the lengths of the sides?

Review 8 Draw the given triangles. Then decide whether

they are similar or not similar. ΔPQR with 16, 8, 18 and ΔXYZ with 9,8,4

418

88

916

P

Q

R

X

Y

Z

16

8 18

9

8 4

418

88

916

ΔPQR is not similar to ΔXYZ

Review 9

Find the value of the variable. 33 q

21 17.5

Review 10 If ΔABC ~ ΔXYZ, AB = 6, and XY = 4, what is

the scale factor of the triangle?

A B

C

X Y

Z

6

4

Scale Factor 6 : 4 3 : 2

Review 11

328 jj83

12

x

32123

y 3

61134

s

328 jj

Review 12

Define scale factor, geometric mean and ratio.Provide an example for each term.

Review 13

XWVSTU

S T

UXW

V

What is the measure of <X?What is the measure of <W?What is the measure of <T?What is the measure of <U?Which side is congruent to TU?

65°

20°

Review 14

Solve the proportion

xx

37

139

xx

14

159

9x = 13(37 – x) 9x = 481 – 13x+13x +13x 22x = 481 x = 21.9

9x = 15 ( 14 – x )9x = 210 – 15x 24x = 210 x = 8.75