Ratios, Proportions, AND Similar Figures 8.1-8.2 Today’s Goal(s): To write ratios and solve proportions. To identify and apply similar polygons.

Ratios A ratio is a comparison of two quantities. The ratio of a to b can be written 3 ways: when b  0

Proportions A proportion is an equation stating two ratios are equivalent. a : b = c : d Read: “a is to b as c is to d”

Do you remember… Means and Extremes Cross-Product Property The cross products of a proportion are EQUAL. ad = bc

Solve each proportion using the cross-product property. a.) b.) c.)

Similar Figures have the same shape, but not necessarily the same size. New symbol: ~ means “is similar to” corresponding angles are congruent (  ) corresponding sides are proportional.

Similarity Ratio The ratio of the lengths of corresponding sides. Determine whether the triangles are similar. If they are, write a similarity statement and give the similarity ratio.

LMNO ~ QRST Find x & write the similarity ratio.

8.2 Extra Examples

 You try this one on your own! 8.2 Extra Examples  You try this one on your own!

Scale Drawings You use proportions all the time in scale drawings. In scale drawings, the scale compares each length in the drawing to the actual length. Example: Suppose you want to make a scale drawing with a scale of 1 in. = 4 ft. What are the dimensions of a 14 ft. by 10 ft. room?

Ex.2 cont…

Benchmark Review Write each in simplest radical form: a.) b.) c.) d.)

Similarity in Right Triangles Toolkit #8.4 Today’s Goal(s): To find and use relationships in similar right triangles.

Page 438 49. <E 55. W(-b,c) Z(-b,-c) 50. <P 56. W(-b,c) Z(-a,0) 51. <Y 57. 6<x<24 52. ZY 53. EZ 54. YZ

Geometric Mean The geometric mean of two positive numbers a and b is the positive number x such that therefore

Find the geometric mean of 4 and 18.

Fill in the angles 50

Now use letters y x

Stop and Think!! How many similar right triangles are formed when you drop a “height” (altitude)? x z y

Take Two Triangles and write the proportion c a b d e c a a b d d+e

Take the “big” with the right.

Big with the left

Take the Left with the Right

Right Triangle “Car” Problem (Understanding the set-up)  HOME RHS

Right Triangle “Car” Problem Time to drive…

Examples: #1

Examples: #2

Examples: #3

Examples: #4

Let’s practice finding the geometric mean of a pair of numbers!  Your answer MUST be in SIMPLEST RADICAL FORM! 4 and 9 4 and 10 5 and 125 7 and 9 x = 6 x = 210 x = 25 x = 37

You Try #1 Solve for x. x = 9

You Try #2 Solve for x. x = 63

You Try #3 Solve for x. x = 12

You Try #4 Solve for x. x = 10

You Try #5 Solve for x. x = 60

You Try #6 Solve for x. x = 20

Cool Down Find the average of 80 and 90. Find the arithmetic mean of 80, 90, 100. Find the geometric mean of 12 and 3. Draw, label and write the geometric mean proportion for x, y, and z.