Ratios, Proportions, AND Similar Figures 8.1-8.2 Todays Goal(s): 1.To write ratios and solve...
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Transcript of Ratios, Proportions, AND Similar Figures 8.1-8.2 Todays Goal(s): 1.To write ratios and solve...
Ratios, Proportions, AND Similar Figures
8.1-8.2Today’s Goal(s):1. To write ratios and solve
proportions.2. 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
b
ababa : " to"
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”
dc
ba
Do you remember…Means and Extremes
ad = bc
Cross-Product Property The cross products of a proportion are EQUAL.
Solve each proportion using the cross-product property.
a.) b.) c.)
n
21
12
9
7
35
4
15
y
y
10
14
2 xx
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 RatioThe 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 ~ QRSTFind x & write the similarity
ratio.
8.2 Extra Examples
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.)
9 12 45 96
Similarity in Right Triangles
Toolkit #8.4
Today’s Goal(s):1. 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
b
x
x
a abx 2
therefore
abx
Find the geometric mean of 4 and 18.
b
x
x
a
Fill in the angles
50
Now use letters
x
y
Stop and Think!!How many similar right triangles are formed when you drop a “height” (altitude)?
xy
z
Take Two Triangles and write the proportion
ab
c
d e
ab
c
d+ed
a
Take the “big” with the right.
Big with the left
Take the Left with the Right
Right Triangle “Car” Problem(Understanding the set-up)
RHS
HOME
Right Triangle “Car” ProblemTime 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 = 25
x = 210
x = 37
You Try #1Solve for x. x = 9
You Try #2Solve for x. x = 63
You Try #3Solve for x. x = 12
You Try #4Solve for x. x = 10
You Try #5Solve for x. x = 60
You Try #6Solve 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.