Download Unit 5: Similarity

Unit 5: Similarity
By Sara Stanley & Leon Chen
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Polygons
Regular Polygons
●
3 or more segments that intersect
exactly two sides; one per endpoint
●
Can be convex - No sides contain a
point inside a polygon
●
Can be concave - Sides contain
point(s) inside polygon
Diagonal - Segment that
joins two non-collinear points
Center of polygon
Apothem - Distance from
center to side of polygon
A Regular Polygon Must Be:
●
Equilateral - All sides congruent
●
Equiangular - All angles congruent
●
Cannot be concave because the interior angles will
not be the same
Connection: Regular polygons are found a lot in Unit
11, where we found area and used them to find
volume of prisms.
Polygon Formulas
N is the number of sides in a polygon; Y is the measure of each exterior
angle
Sum of the interior angles of a polygon = (N
- 2) x 180°
The number of diagonals in a polygon = 1/2 x N(N-3)
N x Y = 360, 360/Y = N, 360/N = Y
Connection: When finding the
perimeter of regular polygons, you also
use (n) as the variable for number of
sides.
Find the measure of an exterior angle
of a dodecagon (12).
360/12 = 30°
Common Mistake:
Be aware when they ask for the SUM
of the measures of the exterior
angles. ALWAYS 360° READ
CAREFULLY
N(180 - 180(N-2)/N)
180N - (180N - 360)
360°
Ratios and Proportions
Find the proportion of:
•
A ratio is a comparison of 2 numbers using division
1
•
Ratio can be expressed as A
to B, A/B, or A:B
•
A proportion is an equation that states that 2 ratios are equal
For example: 4
8
7
14
Connection: We also use ratios when talking about
trigonometry. For example, cosine is the ratio of
the lengths of the adjacent side to the hypotenuse.
x-4
x+3
4x 4x - 18 = x2 - x -12
18
x2 -5x + 6 = 0
x = 2, 3 ----->
Proportion is 1/5, 1/6
Common Mistake:
BE SURE TO PLUG IN BOTH X’s
Remember to check both answers
in a quadratic problem
Similar Triangles
Triangles that have the same angle measurements and the same ratio of side
lengths all the way around
4
45°
5
8
45°
10
65°
70°
65°
14
7
Common Mistake: When trying
to identify if two triangles are
similar, sometimes two side
lengths have the same ratio, but
the last one does not. To avoid
this, remember the ways to
prove similarity: SAS, SSS, AA
No Choice Theorem
40°
40°
60°
60°
If two angles of one
triangle are equal to
two angles of another
triangle then then the
third angle of both
triangles must be
congruent.
Angle Bisector Theorem
If a ray bisects an angle of a triangle, it divides
the opposite side into segments that are
proportional to the adjacent sides.
Given AC bisects ∠BAD, you can say
BC ≌ AB
CD
AD
B
C
D
A
Side Splitter Theorem
If a line is parallel to one side of a triangle and intersects the other
two sides, then it divides those two sides proportionally.
C
Given BD ∥ AE, you can prove that
CB ≌ CD
CA
CE
!
Solve for AE if DE = x, CE = 16, CA = 2x, BA = 8 and BD = 12
B
D
16 - x
16
2x - 8
2x
32x - 2x2 = 32x - 128
x2 - 64 = 0 -------> x = 8
Ratio is now ½ ----> AE is 24
A
E
Real Life Usage
At the same time, many geometric theorems are used in
engineering tasks, one example being this bike. Using the Angle
Bisector Theorem, we could find the exact length of metal needed
to construct that part of the bike.
Similar figures along with ratios and proportions play a vital role
in engineering and architecture. Imagine designing a scale model
of a bridge. To understand each angle and proportional length of
the parts of the bridge is vital to the actual structure.
As a fashion/fabric designer, working with shapes and patterns is
common. It is important to recognize certain properties of
regular polygons. For example, finding the perimeter of a shape,
finding the amount of area in a piece of fabric, etc.
COMMUNICATION
we shall be okay
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