Download 9-Similarity File

Similarity
Unit Essential Question:
How are the ratios of similar figures
used to find missing parts?
Essential Question
How do you use ratios and proportions to
solve problems?
Ratios & Proportions

A ratio is a comparison between two
quantities:

a:b
a to b
What is a ratio?
a/b

How big is the actual bedroom?
Scale Drawings
Does the poster need to be cropped?

A proportion is a statement that two
ratios are equal: 15 10
45


30
An extended proportion:
36 18 2


48 24 3
Proportions
Properties
Examples

These trains are all similar:
What is SIMILARITY?
Essential Question:
When are polygons similar?
Similar Polygons

Two polygons are SIMILAR (~) if:
◦ Corresponding Angles are CONGRUENT
◦ Corresponding Sides are PROPORTIONAL
 The ratio between the sides is called the
SIMILARITY RATIO

ABCD ~ EFGH
◦ m<B =
◦ m<E =
Similarity

Determine if the triangles
are similar. If so, write the
similarity statement and the
similarity ratio.
Example

LMNO ~ QRST

Find x

Find SR
Using Similar Triangles

A Golden Rectangle is a rectangle that can
be split into a Square and a Similar
Rectangle:

In a Golden Rectangle, the
ratio of the length to the
width is the Golden Ratio,
approximately 1.618:1
Golden Rectangle
Sufficient Conditions?
Essential Question:
How do you prove triangles are similar?
Proving Triangles Similar
What are sufficient conditions for
Similarity?

If two angles are congruent, then...
AA~ Postulate

Prove the triangles are similar:
Using AA~
SAS~

Create an extended proportion of 3 sides.
SSS~
Show that the triangles
are similar.
 Write a similarity
statement.

Example

Prove these triangles are similar:
Proving ~
Show the triangles are similar.
 Then find DE.

Applying ~
Using INDIRECT MEASUREMENT
How tall is the cactus?

If m<1 = 65°, then
find the measure of
every numbered angle:

What can you conclude
about the three
triangles?
Consider this…
Essential Question:
What special ratios can be formed using segments
within triangles?
Similarity in Right Triangles

Explanation:
Altitude to Hypotenuse

Proportions with the same number in the
numerator and denominator form
geometric means: 3
x
x

48
x  3  48
2

So,

How is this similar to Arithmetic Means
(averages)?
Geometric Means

Find the Geometric Means of:

4 and 16

5 and 45
Finding Geometric Means
Applying the Corollaries
Small, Medium, Large
Opener
Essential Question:
What special ratios can be formed using segments
within triangles?
Proportions in Triangles

For example:
Side-Splitter Theorem

Similar Triangles:
Compare:

Side-Splitter:
Corollary to Side-Splitter:
Sailing
Triangle-Angle-Bisector
Examples:
Essential Question:
How does a change in the linear dimension of a
figure affect the perimeter, circumference, and
area of a figure?
Perimeters & Areas of Similar Polygons
Perimeter & Area Ratios

Similarity Ratio:

Perimeter Ratio:

Area Ratio:
Perimeter & Area Ratios
Similar Figures
Real-World
Finding Ratios