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Unit 5 Chapter 6
Similarity
By: Devin Zhang and Natalie Hodge
Key Concepts
●
Third Angles Theorem- all angles in a triangle add
up to 180 degrees
● Polygon Formulas- finding areas and side
measurements of polygons
● Triangle Similarity- using the fact that two
triangles are similar, scale factor, and proportions to
find the lengths of the sides of triangles
● Regular Polygons- solving for side lengths and
other things for regular polygons
More Key Concepts
● Proportionality Theorems●
setting up
proportions using theorems that involve parallel lines
and equal line segments
Similar Polygons- using similar polygons to create
proportions and solve for side lengths
● Ratios, Proportionalities, and Geometric
mean- using ratios and proportions to solve problems
and finding the geometric mean
Example #1
Pentagon ABCDE is a regular polygon. BC=5x-4
and DE=2x+11. Find the length of AB.
A
E
B
2x+11
5x-4
D
C
Solution #1
First you have to find x
2x+11=5x-4
1)Since all sides of a regular polygon are the same
length then the two expressions for the side lengths are equal
15=3x
simplify the equation
2)Then you have to
x=5
3)Next, to isolate x you have to divide everything by 3
2(5)+11=AB
expressions
4)After that, you plug the x value into one of the
Example #2
6 is the geometric mean between 12 and what other
number?
Solution: 6=√12x
36=12x
3=x
1. To solve geometric mean, you have to multiply your
two numbers together, then square root them, and
in this case when they are square rooted, they equal 6.
2. To cancel the square root, you must multiply the 6 by
power of 2.
3. By dividing 36 by 12, you end up with the variable X
equaling 3 for your answer.
Example #3
Given the diagram solve for x
3x
2x
18
4x
Solution #3
Solution:
4x
2x
18
=
3x
12x²=36x
12x²-36x=0
12x(x-3)=0
x=3
1) Set up proportion
2) Cross multiply the proportions
3) Subtract 36x to the other side in
order to factor
4) Factor
Common Mistakes
● One common mistake of this unit was that when we
set up proportions, we often set them up wrong and
we end up multiplying the wrong numbers together.
● Another common mistake on the test was that we
forgot to reduce scale factors. This would cause the
whole problem to be off and an incorrect answer.
● Another common mistake of this unit is that we would
have the ratios flipped or wrote it in the wrong unit
making the answer incorrect
Connection #1
In 7a we used the altitude and leg rule which basically is a
proportion between parts of a triangle like the
proportions we set up to solve for x which was a side of
the two triangles.
Leg Rule:
Altitude Rule:
D
A
B
C
Connection #2
● Proportionality in unit 6 is similar to solving
a frustum from unit 12 because in order to
find the volume and surface area of a
frustum, you need the total lateral height
and actual height which can only be solved
through using proportions.
Connection #3
● In unit 5 we learned about triangle sum
theorem and how the interior angles always
adds up to 180 degrees. In unit 8 we
learned the formula (n-2)x180, n
representing the number of angles in the
polygon, these are both ways to find the
interior angle measures of a triangle.
Real Life Usage
●
An environmental scientist may need to find the number of trees in a
150 acre area. If the trees are evenly dispersed then the scientist can
count the number of trees in two acres and create a proportion to
estimate the total number of trees in the 150 acre area such as:
x=total number of trees
x
number of trees in two acres
150
2
=
Real Life Usage #2
● If an architect was building a model for a
skyscraper and he needed to figure out how
tall he needs to build the building he could
use a scale factor and ratios to find out how
tall the building needs to be.
Conclusion
● Using similar figures to create proportions
and ratios that help to solve for x, side
lengths, and geometric mean.