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Sec. 8.1/8.2 Ratios and Proportions
ratio – comparison of two numbers usually in fraction
form where the denominator is not zero
a/b
a to b a:b
proportion – two ratios that are set equal to each other
geometric mean – In the following proportion x is the
geometric mean of a and b.
Properties of Proportions
1. The product of the means equals the product of the
extremes.
2. If ratios are equal, then their reciprocals are equal.
3. If
a c

b d
, then
a b

c d
4. If
a c

b d
, then
ab cd

b
d
Examples :
1. Simplify
12cm
4m
2. Simplify
6 ft
8in
True or False
p r
3. If 6  10 , then
4. If
a c
 ,
3 4
then
p 3

r 5
.
a3 c3

3
4
.
5. The perimeter of a rectangle is 60 cm. The ratio of
AB to BC is 3 : 2. Find the length and the width of
the rectangle.
6. The measures of the angles of a triangle is 1 : 2 : 3,
find the measures of the angles.
4 5
7. Solve the proportion : x  7
8. Solve the proportion :
3
2

y2 y
9. 5. A scale model of the Titanic is 107.5 inches long
by 11.25 inches wide. The Titanic was 882.75 feet
long…how wide was it?
Sec. 8.3 Similar Polygons
Similar polygons – polygons with corresponding
angles congruent and corresponding sides proportional.
scale factor – the ratio of the corresponding sides
What is the scale factor for the similar polygons?
What is the ratio of their perimeters?
Thm. 8.1
If two polygons are similar, then the ratio of their
perimeters is equal to the ratios of their corresponding
side lengths.
Example :
What is the scale factor of the similar polygons?
Find z.
Sec. 8.4 Similar Triangles
similar triangles – triangles that have congruent,
corresponding angles and proportional, corresponding
sides.
Postulate 25 Angle-Angle(AA) Similarity Postulate
If two angles of one triangle are congruent to two
angles of another triangle, then the triangles are similar.
Example :
1. In the diagram, ∆BTW~∆ETC.
a. Write the statement of proportionality.
b. Find mTEC
c. Find ET and BE
2. Find the length of the altitude segment QS.
Sec. 8.5 Proving Triangles Are Similar
Side-Side-Side (SSS) Similarity Theorem
If the lengths of the corresponding sides of two
triangles are proportional, then the triangles are similar.
Side-Angle-Side (SAS) Similarity Theorem
If the angle of one triangle is congruent to an angle
of another triangle and the sides that include these
angles are proportional, then the triangles are similar.
Examples :
1. Which two triangles are similar?
2. What is the distance across the river?
Sec. 8.6 Proportions and Similar Triangles
Thm. 8.4 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.
Thm. 8.5 Converse of the Triangle Prop. Thm.
If a line divides two sides of the triangle
proportionally, then it is parallel to the third side.
Thm. 8.6
If three parallel lines intersect two transversals, then
they divide the transversals proportionally.
Thm. 8.7
If a ray bisects an angle of triangle, then it divides
the opposite side into segments whose lengths are
proportional to the lengths of the other two sides.
Sec. 8.7 Dilations
Dilations – a non-rigid transformation in which the preimage is similar to the image. The following properties
are true :
Reduction : 0 < k < 1
Enlargement : k > 1