In this lesson you’ll explore the basics of similar polygon and
congruent polygon, concept of scale factor calculating it for an enlargement or reduction. From there move forward to write similarity statements for two similar polygons, explore how to determine if two figures are similar and calculate missing side lengths for similar polygons. The contents will be presented by the instructor in own handwriting and using video and with the help of several examples. The leaning can be applied to solve the real world problems and help for Geometry home work.
(More text below video...)
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(Continued from above)
Recall earlier learning, you know that figures that have the same size and shape are congruent figures. But figures that have the same shape but not necessarily the same size are similar figures.
Similar figures may be as enlargements or reductions of one another with no distortion. Likewise similar Polygons are polygons containing vertices that can be paired such that the corresponding angles are congruent and the corresponding sides are in proportion. The ratio of the lengths of two corresponding sides of similar polygons is called scale factor.
For example: the pentagons shown in the figure below are similar; but the rectangles are not
similar since it can not be enlarged or reduced to fit exactly one over the other.
The relationships can be reproduced as below: “Two polygons are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional”.
Notice that even if the corresponding sides are proportional, but corresponding angles are not congruent they are not similar. It is essential that for two figures to be similar, both conditions-
proportional sides and congruent angles—must hold.
Example 1:
Given- Determine whether parallelogram mnop is similar to parallelogram ABCD.
Step 1: mn
= mA
i.e. each angle is 60.
Step 2: Use angle properties of the parallelogram,
Using angle properties of parallelograms, mm
= mD
= 120°.
Then mp
= mC
= 60°,
and mo
= mB
= 120° i.e. the corresponding angles are congruent ().
Step 3: Now about sides, mn/DA = 9/12 = 3/4, and no/AB = 12/18 = 2/3. Thus the corresponding sides are not proportional.
Therefore, the parallelograms are not similar, as the final answer.
Example 2: Given- find a, if the two pentagons are similar.
Step 1: The two pentagons in the figure on right are similar i.e. the sides are
proportional.
Step 2: mq/AE =qp/ED, Thus
2/a = 4/(a+5).
Simplifying it gives- a = 5 units.
Thus a=2, as the final answer.
Example 3: Given- mnop and ABCD are similar figures. Find
the length of .
Step 1: Figure mnop and ABCD are similar (Given). Thus the sides are proportional.
Step 2: 8/14 = 6/a
Simplification gives a= 10.5 units.
Thus length of BC is 10.5, as the final answer.
Example 4: Given- polygon ABCD and mnop. Calculate the scale factor of the two polygons shown in the figure.
Step 1: Study the given figure figures and recall the properties.
A
is congruent to
E,
B
is congruent to
F,
and
C
is congruent to
G,
&
D
is congruent to
H
Step 2: AB/mn = BC/no= CD/op = AD/mp (properties of similar polygons)
The scale factor of polygon ABCD to polygon mnop is 16/8 = 2, as the final answer.
Example 5: Given - the figure (below). Determine if these are similar. If so, write a similarity statement.
Step 1: Check if the corresponding angles are , m
A,
o
C,
n
B.
Step 2: Check if the corresponding sides are proportional.
mn/AB=3/7.5=1/2.5
no/BC=5/12.5=1/2.5, and
om/CA=4/10=1/2.5
Since the three angles are congruent and sides are proportional, therefore, mno ~ ABC,
is the final answer.
The video above will explain more in detail about Similar Polygons,and
how to apply the concepts in solving real-world problems. This is explained with the help of several examples and done watching video. This helps you to deal with solving problems and help doing the Geometry home work.
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n
= m
.
).
mno ~ 