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SIMILARITIES
Similar polygons are polygons having the same angles and proportions,
though of different size. Figures that have a same shape, but do not have
the same size are said to be similar.
Congruent is the figure identical in form. i.e. same size and shape.
Congruent angles=angles that have the same measure.
Congruent polygons=polygons that are same size and shape.
Congruent figures are also similar, but similar figures are not necessarily
congruent.
Some shapes are always proportional to each other:
-all circles are similar to each other (all proportional to each other)
-all squares are similar to each other (90° angles and same proportions)
Example 1:
These are similar polygons. To prove this statement, you need to show their
proportional to each other:
AB:ab = 9:6
BD:bd = 3:2
=3:2
And therefore AB:ab = BD:bd
All the angles in both rectangles are equal (90°)
The fact that we get same scale factor means that they are same the
proportions, which agrees with them being similar.
Example 2:
These are similar polygons as well. To prove this statement, you need to show
they are proportional to each other:
AB:ab = 6:3
= 2:1
BC:bc = 4:2
= 2:1
CD:cd = 4:2
DA:da = 8:4
= 2:1
= 2:1
Polygon ABCD is an enlargement of polygon abcd. The lengths have doubled,
but the angles stayed the same. Thus, they are similar polygons.
Note: for any pair of similar figures corresponding sides are in the same
ratio and corresponding angles are equal.
Generally, to prove two polygons are similar, you must prove that all
corresponding angles are equal and all ratios of corresponding sides are
equal, but for triangle there’s no need.
Special similarity rules for triangles:
-If two angles of one triangle are congruent to two angles of another triangle,
then the triangles are similar.
-If all pairs of corresponding sides of two triangles are proportional, then the
triangles are similar.
-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.
Example 3:
These two triangles are similar to each other because all corresponding angles
are equal. To find y:
5:10 = 1:2 = 4:8
Therefore, 1:2 = 3:y
= 3:6 (by cross multiplication)
y=6
Example 4:
In this figure, there are two triangles: triangle PST and triangle PQR. Line ST and
QR are parallel.
Triangle PST is similar to triangle PQR because:
∠SPT = ∠QPR
∠PST = ∠PQR (corresponding angles)
∠STP = ∠QRT (corresponding angles)
PS:PQ = PT:PR
PS:PQ = 4:10
=2:5
PT:PR = x:(12+x)
2:5 = x:(12+x)
x=8
PS:SQ = PT:TQ because:
4:6 = 2:3 = 8:12
Note: 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.
Example 5:
First of all you need to prove triangle ABC and triangle ADE are similar.
BC//DE
So, ∠BAC = ∠DAE (vertically opposite angle)
∠ABC = ∠AED (alternate angles)
∠ACB = ∠ADE (alternate angles)
All 3 corresponding angles are equal, so triangle ABC~ triangle ADE
AB:AE = 7:21
=1:3
AC:AD = x:9
1:3 = x:9
x = 3cm
Perimeters of the similar triangles
The ratios of corresponding sides are:
5:10 = 1:2
4:8 = 1:2
3:6 = 1:2
Perimeter of triangle abc: perimeter of triangle ABC
12:24 = 1:2
Note: therefore, the scale factor of two similar triangles is same as the
ratio of their perimeter.
Area of the similar triangles
Scale factor of this two similar triangles are 1:2
4:8 = 1:2
3:6 = 1:2
5:10 = 1:2
Area of triangle abc = 1/2 x 3cm x 4cm
= 6cm2
Area of triangle ABC = 1/2 x 6cm x 8cm
= 24cm2
6:24 = 1:4
Note: therefore one can say that when the scale factor of similar triangle is
a:b, the ratio of their area will be a2:b2.
Note: thus, one also could say that when the ratio of perimeter is a:b, the
ratio of the area will be a2:b2.
Example 6:
Triangle abc ~ triangle ABC.Find their ratio of the area
ab:AB = 2:6
=1:3
bc:BC = 3:k
k = 9 (cross multiplication)
area of triangle abc = 1/2 x 3 x 2
= 3cm2
area of triangle ABC = 1/2 x 9 x 6
= 27cm2
Ratio of the area = 3:27
=1:9
=12:32
= (1:3)2
Similarity of 3Dshapes
3Dshapes are considered to be similar under these conditions:
-when all corresponding angles are equal
-when all corresponding sides are proportional to each other.
Surface area of 3D shapes
Ratio of length = a:b
Surface area of large cube = 6a 2.
Surface area of small cube = 6b 2.
6a2:6b2 = a2:b2
Volume of 3D shapes
Ratio of length = a:b
Volume of small cube = b3
Volume of large cube = a3
a3:b3
Collected rules
For any type of similar shapes following is true:
When scale factor or ratio of length is a:b:
Ratio of perimeter = a:b
Ratio of area = a2:b2
Ratio of surface area = a2:b2
Ratio of volume = a3:b3