Download Similar Polygons: Two polygons containing vertices that can

Similar Polygons: Two polygons containing vertices that can be paired so
that the corresponding angles are congruent and the corresponding sides are
in proportion. The symbol for similarity is ~.
When 2 polygons are similar, the following conditions must be fulfilled:
1) Corresponding angles are equal
2) The ratios of pairs of corresponding sides must all be equal.
3) Corresponding diagonals are in the same proportion.
4) The ratio of the areas of the two polygons is the square of the ratio of the
sides.
Similar polygons are polygons for which all corresponding angles are
congruent and all corresponding sides are proportional. Example:
If any of the above conditions do not occur, then the polygons are not
similar.
Many times you will be asked to find the measures of angles and
sides of figures. Similar polygons can help you out.
1. Problem: Find the value of x, y, and
The measure of angle P.
Solution: To find the value of x and y,
write proportions involving corresponding
sides. Then use cross products to solve.
4 x
-=6 9
6x = 36
x=6
4 7
-=6 y
4y = 42
y = 10.5
To find angle P, note that angle P
and angle S are corresponding angles.
By definition of similar polygons,
angle P = angle S = 86o.
In general to prove that polygons are similar, you have to show that all pairs
of corresponding angles are equal and that all ratios of pairs of
corresponding sides are equal too. However this is not necessary to show
the similarity of triangles.
SIMILAR TRIANGLES.
The triangle, geometry's pet shape :-) , has a couple of special rules
dealing with similarity. They are outlined below.
1. Angle-Angle Similarity - If two angles of one triangle are congruent to
two angles of another triangle, then the triangles are similar.
1. Problem: Prove triangle ABE is similar
to triangle CDE.
Solution: Angle A and angle C are congruent (this
information is given in the figure).
Angle AEB and angle CED are
congruent because vertical angles are
congruent.
Triangle ABE and triangle CDE are similar
by Angle-Angle.
2. Side-Side-Side Similarity - If all pairs of corresponding sides of two
triangles are proportional, then the triangles are similar.
3. Side-Angle-Side Similarity - If one angle of a triangle is congruent to one
angle of another triangle and the sides that include those angles are
proportional, then the two triangles are similar.
2. Problem: Are the triangles shown in
the figure similar?
Solution: Find the ratios of the
corresponding sides.
UV 9 3
-- = -- = KL 12 4
VW 15 3
-- = -- = LM 20 4
The sides that include angle V
and angle L are proportional.
Angle V and angle L are
congruent (the information is given in the figure).
Triangle UVS and triangle KLM are similar by Side-Angle-Side.
What do parallel lines and triangles have to do with similar polygons? Well,
you can create similar triangles by drawing a segment parallel to one side of
a triangle in the triangle. This is useful when you have to find the value of a
triangle's side (or, in a really scary case, only part of the value of a side).
The theorem that lets us do that says if a segment is parallel to one side of a
triangle and intersects the other sides in two points, then the triangle
formed is similar to the original triangle. Also, when you put a parallel line
in a triangle, as the theorem above describes, the sides are divided
proportionally.
1.
Problem: Find PT and PR
Solution: 4 x
- = -7 12
because the sides are divided
proportionally when you draw a
parallel line to another side.
Cross products
7x = 48
x = 48/7
PT = 48/7
PR = 12 + 48/7 = 132/7
Scale factor: The ratio of the lengths of two corresponding sides of similar
polygons
/ A is congruent to / E
/ B is congruent to / F
/ C is congruent to / G
/ D is congruent to / H
AB/EF = BC/FG= CD/GH = AD/EH
The scale factor of polygon ABCD to polygon EFGH is 10/20 or ½
PERIMETERS AND AREAS OF SIMILAR TRIANGLES.
When two triangles are similar, the reduced ratio of any two corresponding sides is
called the scale factor of the similar triangles. In Figure 1 , Δ ABC‫ ׽‬Δ DEF.
Figure 1Similar triangles whose scale factor is 2 : 1.
The ratios of corresponding sides are 6/3, 8/4, 10/5. These all reduce to
2/1. It is then said that the scale factor of these two similar triangles is 2
: 1.
The perimeter of Δ ABC is 24 inches, and the perimeter of Δ DEF is 12
inches. When you compare the ratios of the perimeters of these similar
triangles, you also get 2 : 1. This leads to the following theorem.
Theorem 60: If two similar triangles have a scale factor of a : b, then the
ratio of their perimeters is a : b. (PERIMETER)
Example 1: In Figure 2 , Δ ABC‫ ׽‬Δ DEF. Find the perimeter of Δ DEF
Figure 2 Perimeter of similar triangles.
AREA.
Figure 3 shows two similar right triangles whose scale factor is 2 : 3.
Because GH ٣ GI and JK ٣ JL, they can be considered base and height for
each triangle. You can now find the area of each triangle.
Figure 3Finding the areas of similar right triangles whose scale factor is 2 :
3.
Now you can compare the ratio of the areas of these similar triangles.
This leads to the following theorem:
If two similar triangles have a scale factor of a : b, then the ratio of their
areas is a2 : b2.
SIMILARITY OF 3-D FIGURES.
The following rules must be followed for a3-D figure to be similar:
1. Angles are equal
2. The ratios of pairs of corresponding sides must all be equal.
If two similar 3-D shapes have a scale factor of a : b, then the ratio of their surface areas
is a2 : b2.
If two similar 3-D shapes have a scale factor of a: b, then the ratio of their volumes is a3 :
b3