6-1 Using Proportions

•I. Ratios and Proportions

Ratio- comparison oftwo or more quantitiesExample: 3 cats to 5 dogs 3:5 3 to 5 3/5

Proportion: two equal ratios: Example: 3/5 = 6/10

Equality of Cross ProductsA = C if and only if AD=BCB D

A and D are the extremesB and C are the means

II. Examples

• 1. X = 20 2. X+ 3 = 4 4 5 9 3

Solutions:

x • 5 = 20 • 4

5x = 80

x = 16

3(x + 3) = 4•9

3x + 9 = 36

3x = 27 x = 9

• 3. A + 1 = 5 A- 1 6

• 4. Find 32 % of 156.

6(A + 1) = 5(A – 1)

6A + 6 = 5A – 5

A + 6 = -5 A = -11

_is_ = _%_ of 100

_x_ = _32_156 100

100x = 4992 x = 49.92

• 5. Write a proportion to find x.

X +2 4

20 16

X + 2 = _4_ 20 16

16( X + 2) = 4(20)

16X + 32 = 80

16X = 48 X = 3

a:b:c represents a:b b:c a:c

6. The ratio of the measures of three sides of a triangle is 8:7:5 and the perimeter of the triangle is 240 cm. Find the measure of each side of the triangle

8x + 7x + 5x = 240

20x = 240

X = 12

8(12) = 96

7(12) = 84

5(12) = 60

Complete P. 341: 6, 7, 9, 11 – 13, 15

6. 1:75 or 1/757. 170:9 or 170/99. X = 1411. Yes12. Yes13. No15. 3x + 4x + 5x = 72 12x = 72 x = 6 3(6) = 18

4(6) = 24 5(6) = 30

6-2 Exploring Similar Polygons

• I. Similar polygons– All corresponding sides must be

proportional and all corresponding angles must be congruent

– Symbol is ~

II. Scale factors and dilations

• Scale factor: ratio of two corresponding lengths

• Dilation: a transformation that reduces or enlarges

ENLARGEMENT

REDUCTION

Ratio of the sides (scale factor) a:b

Ratio of the perimeters a:b(same as scale factor)

Ratio of the Areas a²:b²(remember: area is always squared)

III. Examples

• 1. Find the scale factor for the similar polygons

8

6 6

8

20

15

20

The scale factor of the smaller to the larger is 6/15 or 8/20 (both = 2/5)

The scale factor of the larger to the smaller is 15/6 or 20/8 (both = 5/2)

15

2. Polygon RSTUV and polygon ABCDE are similar.

R

S T

U

VA

B C

D

E

a. Find the scale factor of RSTUV to ABCDE

b. Find the value of xc. Find the value of y

18/4 = 9/2

18/4 = x/3

18(3) = 4x 13.5 = x

18/4 = (y+2)/5 18(5) = 4(y+2)

90 = 4y + 8 82 = 4y y = 20.5

x

3

18

4

y+2

5

• 3. Perform a dilation of 3 on the square with coordinates:

• D(0,0) E (5,0) F(0,5) G (5,5)

(0,0) (5,0)

(0,5) (5,5)

(0,0) (15, 0)

(0,15) (15,15)

Complete p.349 and p.350: 1, 2, 4, 8 - 11

1. A. Yes, all corresponding angles are congruent and all

corresponding sides are proportional (1:1) B. No, sides are proportional not necessarily congruent

2. They both could be right: Larger to smaller or smaller to larger4. The corresponding angles must be congruent8. Sometimes. The sides might not be proportional and the acute angles might not be congruent9. Always: if figures are congruent they are similar10. A. 6/9 = 2/3 11. (0,0) (9,0) (0,12) B. 6/9 = 12/x 6/9 = 14/y yes, scale factor of x = 18 in. Y = 21 in. old to new is 1/3

C. 63 in. D. 2/3

6-3 Identifying Similar Triangles

• I. Methods that show if triangles are similar

• AA Similarity

• SSS Similarity

• SAS Similarity

II. AA Similarity

• If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 

III. SSS Similarity

• If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. 

IV. SAS Similarity

• If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. 

V. Examples

• 1. Find CB

C

D

E

A

B

AB // DE

6

3

2

Since the lines are //, the twosets of corresponding angles are congruent and the trianglesare similar by AA.

6

3

8

x

6/8 = 3/x

6x = 8(3)

x = 4 so CB = 4

2. Find the value of x, AE, and ED

A B

C D

E

4

8

3x + 4

X + 12

AB // CD

Since the lines are parallel,The two pair of alternate interiorangles are congruent and thetriangles are similar byAA.

So, 4/8 = (3x + 4)/(x + 12) Then 4(x + 12) = 8(3x + 4)4x + 48 = 24x + 32 x = 4/5 and AE = 3(4/5) + 4 = 6.4 ED = 4/5 + 12 = 12.8

Complete p. 357: 2, 4, 6-11

2. Only one; you know the right angles are congruent and if you know one pair of acute angles are congruent then the third angles are automatically congruent. 4. They are both correct. In both cases the cross product is the same6. Yes, they are similar by AA7. Not enough information8. True, SSS similarity9. Triangle AEC and Triangle BDC by AA10. AA similarity 3/12 = x/(x+12) x = 4 3/12 = y/20 y = 511. SAS similarity 2/4 = x/9 x = 4.5

6-4 Parallel Lines and Proportional Parts

• I. Triangle proportions

• If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths.

This line makes a smaller similar triangle to the outer big triangle

• The converse is also true- if you have two similar triangles, a line intersecting is parallel.

II. Midpoints

• A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side.

III. Many transversals

• If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

IV. Examples

1. You want to plant a row of trees along a slope as shown. Find the indicated lengths.

Assume all of the vertical lines are parallel.

10 12 10 16 18

X

Y

XY = 88a

b

c

d

e

a/10 = 88/66 a = 13 1/3

b/12 = 88/66 b = 16

c/10 = 88/66 c = 13 1/3

d/16 = 88/66 d = 21 1/3

e/18 = 88/66 e = 24

2. Complete the following statements:

a. a/b = c/___

b. c/e = d/___

c. a/e = b/___ a

b

c

d

e

f

Answers:

a. db. fc. f

3. Complete the following:

a. PQ/PR = QS/___

b. PQ/QR = PS/___

c. PR/PT = PQ/___

P

Q R

S T

Answers

a. RTb. STc. PS

4. Find the value of x and y

x + 2

3x - 9

y+ 2

3y -8

Answers:

2(3x – 9) = x + 2 6x – 18 = x + 2 5x = 20 x = 4

y + 2 = 3y – 8 10 = 2y 5 = y

Complete page 366: 6 - 12

6. A. LT B. RL7. A. True B. False, RS = 168. x = 2 and y = 129. Yes10. No, must have DG // EF11. Yes12. x = 3 1/3 feet y = 2 2/3 feet z = 2 feet

6-5 Parts of Similar Triangles

• I. Perimeters

• If two triangles are similar, then the perimeters are proportional to the measures of corresponding sides.

II. Altitudes

• Corresponding altitudes are proportional to the measures of the corresponding sides.

III. Angle bisectors

• Corresponding angle bisectors are proportional to the measures of the corresponding sides.

IV. Medians

• Corresponding medians are proportional to the measures of the corresponding sides.

V. Angle bisectorAn angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

Examples:

1. Triangle ABC is similar to triangle DEF. Find the perimeter of triangle ABC.

A

BC

D

E F

40

9

419

Answer:

AB/DE = Perimeter ABC/ Perimeter DEF 9/40 = x/90 x = 20.25

4

x

2. Find EH. Triangle ABC is similar to Triangle DEF.

A G C

B

D H F

E

5

30

Answer: 4/x = 5/30 x = 24

3. Complete the proportion for the similar triangles:

c

f

a b

d e

a. a/d = c/___

b. b/c = e/___

c. a/b = d/___

Answers:a. fb. fc. e

4. Find the value of x and y for the similar triangles.

25

x + 23x - 4

3y - 4

2y + 12

Answers: 2/5 = (x + 2) / (3x – 4) 2(3x – 4) = 5(x + 2) x = 18

2/5 = (3y – 4) / (2y + 12) 2(2y + 12) = 5(3y- 4) y = 4

5. Find the value of x.

1214

3 X

12/3 = 14/x x = 3.5

Complete page 373: 5 - 9

5. AB: angle bisector theorem6. DF: The medians of two similar triangles are proportional to two corresponding sides.7. X = 158. X = 6.759. x/5 = 2/(7 – x) x(7 – x) = 5(2)

7x - x² = 10 0 = x² - 7x + 10

0 = (x – 5) (x – 2) x = 5 or 2

x = 2 isn’t possible because

the leg and

hypotenuse of a

right triangle

are not equal.