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Similarity
Two geometrical objects are called similar if they
both have the same shape. More precisely, one is
congruent to the result of a uniform scaling
(enlarging or shrinking) of the other.
Corresponding sides of similar polygons are in
proportion, and corresponding angles of similar
polygons have the same measure. One can be
obtained from the other by uniformly "stretching"
the same amount on all directions, possibly with
additional rotation and reflection, i.e., both have
the same shape, or one has the same shape as the
mirror image of the other. For example, all circles
are similar to each other, all squares are similar to
each other, and all parabolas are similar to each
other. On the other hand, ellipses are not all
similar to each other, nor are hyperbolas all similar
to each other. If two angles of a triangle are equal
to two angles of another triangle, then the
triangles are similar.
What are similar polygons?
Similar polygons are polygons for which all
corresponding angles are congruent and all
corresponding sides are proportional. Example:
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.
Special Similarity Rules for
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.
Quiz on Similarity of Triangles
1. All congruent triangles are similar, but all similar triangles
are not congruent.
True
False
2. In a triangle ABC, a line is drawn parallel to BC, which cuts
AB at D and AC at E.
The ratio of AD:AB is equal to 3:5.
What is the ratio of DE:BC?
3:4
Impossible to say
3:5
4:5
3. What is the sign which denotes similarity?
~
:=
^
---
4. In a triangle XYZ, a line 'PQ' is drawn parallel to YZ, cutting
XY at P and XZ at Q.
The ratio of XP:XY is 2:3. What is the ratio XQ:QZ?
2:3
2:1
Impossible to say
2:5
5. If a line divides two sides of a triangle in proportion, then it
is parallel to the third side.
True
False
6. The Midpoint Theorem states:
"A line joining the midpoints of any two sides of a triangle is
parallel to the third side and equal to half the length of the
third side."
So, riddle me this: What is the converse of the Midpoint
Theorem?
None of these are the converse.
"If a line is parallel to one side of a triangle and is equal to
half the length of the same side, the given line passes through
the midpoints of the other two sides."
"If a line passes through the midpoint of one side of a
triangle and is parallel to a second side, the given line bisects
the third side."
"If a line passes through the midpoint of one side of a
triangle, and is equal to half the length of a second side, it
bisects the third side."
7. Enough about sides. Let's talk areas.
You have been given two similar triangles, MNO and DEF. You
have to find out the ratios of their areas.
Which of the following formulae can be used to find out the
ratio of the areas of the given triangles.
The ratio of the areas of the triangles is equal to the square
of the corresponding sides
The ratio of the areas of the triangles is equal to the square
of the corresponding altitudes
The ratio of the areas of the triangles is equal to the square
of the corresponding medians
All of these can be used to evaluate the ratio of the areas
of the triangles
8. All equilateral triangles are similar.
True
False
9. A given triangle ABC is isosceles. Angle B and angle C are
the base angles.
From B, a perpendicular 'BD' is drawn to cut AC at point D.
From C, a perpendicular 'CE' is drawn to cut AB at point E.
State which of the following pairs of triangles are similar.
DBC ~ AEC
BDC ~ AEB
BDC ~ CEB
CDB ~ BCA
10. ABC and PQR are two triangles. If the ratios of all the
corresponding sides of the triangles are equal, then ABC ~
PQR.
True
False
Answers to the Quiz
1. The correct answer was true
To define similarity simply, it is a case in which
all the corresponding angles of two or more
triangles are equal, but their sides are not
necessarily equal.
Congruent triangles are triangles which are
identical in every way.
As such, all congruent triangles are similar, but
all similar triangles are not congruent.
2. The correct answer was 3:5
Triangle ADE was similar to triangle ABC.
A striking property of similar triangles is that
the ratio of all their corresponding sides are
equal. For example, if triangle ABC is similar to
triangle PQR, then :AB:PQ=AC:PR=BC:QR
3. The correct answer was ~
ABC~PQR means that triangle ABC is similar to
riangle PQR.
4. The correct answer was 2:1
In order to evaluate this answer, you had to use
the basic proportionality theorem, which states,
“if a line is drawn parallel to a side of a triangle,
it divides the other two sides in proportion.”
For example, in the above mentioned case,
XP:PY-XQ:QZ. Therefore, since the ratio XP:XY
was given, you were required to find out the
ratio of XP:PY.
XP:XY= 2:3
Let XP=2x and XY be 3x
PY=XY-YP= 3x-2x=x
Therefore XP:PY=2x:x=2:1
But XP:PY=XQ:QZ
Therefore, XQ:QZ=2:1
5. The correct answer was true
In ABC, DE is a line which cuts AB at D and AC
at E. AD:DB=AE:EC.
Therefore, DE is parallel to the third side of
ABC, namely BC
6.
The correct answer was “if a line passes
through the midpoint of one side of a
triangle and is parallel to a second side, the
given line bisects the third side.”
7. The correct answer was the ratio of the
area of the triangles is equal to the square
of the corresponding sides
8. The correct answer was true.
9. The correct answer was BDC~CEB
10. The correct answer was true.
Parallel lines & Triangles
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
- = -because the sides are divided
7 12
proportionally when you draw a
parallel line to another side.
7x = 48
x = 48/7
PT = 48/7
Cross products
PR = 12 + 48/7 = 132/7
Congruency
If two shapes are congruent, they are identical in
both shape and size.
Remember: Shapes can be congruent even if one of
them has been rotated or reflected.
The symbol
means 'is congruent to'.
Two triangles are congruent if one of the following
conditions applies:
1. Three sides are the same
The three sides of the first triangle are equal to
the three sides of the second triangle (the SSS
rule: Side Side Side).
2. Two sides and one angle are the same
Two sides of the first triangle are equal to two
sides of the second triangle, and the included angle
is equal (the SAS rule: Side Angle Side).
3. Two angles and one side are the same
Two angles in the first triangle are equal to two
angles in the second triangle, and one (similarly
located) side is equal (the AAS rule: Angle Angle
Side).
4. Two sides in right-angled triangle are the same
In a right-angled triangle, the hypotenuse and one
other side in the first triangle are equal to the
hypotenuse and corresponding side in the second
triangle (the RHS rule: Right-angled, Hypotenuse,
Side).
Question
For each of the following pairs of triangles,
state whether they are congruent. If they are,
give a reason for your answer (SSS, SAS, AAS
or RHS).
Pair 1
Pair 2
Pair 3
Answer
1. Yes. RHS
2. Yes. SSS
3. No. The side of length 7cm is not in the
same position on both triangles. Therefore,
it is not AAS.
Similar areas and volumes
We already know that if two shapes are similar
their corresponding sides are in the same ratio,
and their corresponding angles are equal.
Look at the two cubes below:
The cubes are similar, and the ratio of their
lengths is a:b or
Question
What is the ratio of:
1. the area of their faces?
2. their volumes?
Answer
1. Cube 'a' has a face area of a2
Cube 'b' has a face area of b2
The ratio of their areas is a2 :b2 or
2. Cube 'a' has a volume of a3
Cube 'b' has a volume of b3
The ratio of their volumes is a3 :b3 or
For any pair of similar shapes, the following is
true:
Ratio of lengths = a:b or
Ratio of areas = a2:b2 or
Ratio of volumes = a3:b3 or
Question
These two shapes are similar. What is the
length of x?
Answer
The ratio of the areas is 25:36 (a2:b2)
The ratio of the lengths is a:b Therefore, we
find the square roots of 25 and 36. Ratio of
lengths = 5:6
5:6 = 2:x
So
(multiply both sides by 2) = x
x = 2.4cm
Now you try one.
Question
Two similar pyramids have volumes of 64cm3
and 343cm3. What is the ratio of their surface
areas?
Answer
The answer is 16:49. Here is how to work it
out. Try to fill in the blanks below:
Ratio of volumes = 64:?
To find the ratio of the lengths, we find the
cube roots of 64 and ?
Therefore, the ratio of the lengths is ?:7
To find the ratio of the areas, we square ? and
7
The ratio of the areas is 16:49.
done by farhan
grade 11d