unit 8 similarity

Unit 8 Similarity

Ratios, Proportions, Similar Polygons

Geometry w/Ms. Hernandez

St. Pius X High School

Unit 8 Similarity

• This unit is about similar polygons. Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional.

• Some vocabulary: ratio, proportion, means, extremes, geometric mean, similar polygons, scale factor

8.1 Ratio• If a and b are two quantities that are

measured in the SAME units, then the ratio of a to b is a : b.

• Ratios can be written 3 different ways.– a to b – a : b–

a

b

8.1 Ratio

• You must be able to simplify ratios

• Example 12 12 12 3

4 400 400 100

cm cm

m cm

In this example we used the fact that there are 100 cm in 1 m. So then 4m = 4 x 100cm = 400cm. Next, we reduced 12/400. 12 and 400 both have the factor 4 in common. So we can factor out a 12 from the numerator and denominator.

12 4 3 4 3 3

400 4 100 4 100 100

8.1 Ratio• You must be able to USE ratios

• Example– The perimeter of a rectangle ABCD is 60

centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle.

At first glance it may look like we don’t have enough information to solve this problem. Let’s take apart the given information and see how we can solve it.

8.1 Using Ratios– The perimeter of a rectangle ABCD is 60


perimeter of a rectangle ABCD is 60 centimeters

Since we are working with rectangle ABCD let’s draw it first.

D

AB

C



perimeter of a rectangle ABCD is 60 centimeters. The perimeter of rectangle is found from the formula P = 2l + 2w where l = length and w = width.So we then could write 60cm = 2l + 2w

D

AB

C



ratio of AB:BC is 3:2Remember we can write ratios in different ways, therefore

we can write the ratios as fractions

D

AB

C

3

2

AB

BC



Find the length and width of the rectangle AB represents length in this rectangle and BC represents width in this rectangleSo we can write AB = 3x and BC = 2x

D

AB

C

3

2

AB x

BC x



Putting it all together

D

AB

C

3

2

AB x

BC x60cm = 2(3x) + 2(2x)

60cm = 6x + 4x 60cm = 10x6cm = xL = AB = 3x = 3(6) = 18W = BC = 2x = 2(x) =12

8.1 Using Ratios

• This same technique can be applied in area problems and measures of interior angles of a triangle. In section 8.1 see example #3 in your textbook and problems #29-32 in your textbook.

8.1 Extended Ratios• BC:AC:AB is 3:4:5

Solve for x.So then we write the following ratios

6

x + 2

xA

B

C

3

44

53

5

BC

ACAC

ABBC

AB

And we can use any of these ratios to solve for x.

3 6 3

4 44 4

5 2 53 6 3

5 2 5

BC

AC xAC x

AB xBC

AB x

8.2 Problem Solving in Geometry with Proportions• We can take advantage of the following

properties of proportions to being solving problems in Geometry.

a c a bIf

b d c d = , then =

a c a b c dIf

b d b d

= , then =

8.2 Examples

• In the diagram

AB = AC. Find the length of .

BD CE

30

10x

16

D

A

E

B C

BD

16 30 10

1016 20

10160 20

8

AB AC

BD CE

x

xx

x

8BD x

8.2 More examples

• Use your textbook to complete more examples. Sometimes x is not in the picture and you have to figure out where it goes.

• Try these problems from section 8.2 #23-28

8.2 Geometric Mean

• The geometric mean of two numbers a and b is x and is represented by the following proportion. a x

x b

Find the geometric mean of 3 and 27.

2

2

33 27 81

27

81 9

xx x x

x

x x

8.2 Geometric Mean

• Another way to respsent the geometric mean of two numbers a and b is.

x a b

Find the geometric mean of 3 and 27.

3 27 81 9x x x

unit 8 similarity

Documents