Similar Triangles

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Similar Triangles are the same general shape as each and differ only in size. This is also true for all other groups of similar figures. The figures below that are the same color are all similar.

various similar figures

Note: All circles, of any diameter, are similar figures. The same can be said for all squares, and all equilateral triangles.

Students will typically study Similar Triangles and other similar figures in 8th Grade

Similarity of figures is often discussed long with the concept of Congruence. Congruent figures are not only the same shape (as with similar figures), they are also the same size. You will find more on Congruent figures here along with guidance on the different types of transformations that can be applied to generate congruent or similar figures.

The difference between similarity and congruence is that similar figures have been to been subjected to a dilation (or, in more common language, they have been re-sized, or scaled, or enlarged, or shrunk). You will find more here on dilations and similar figures.

How to Tell if Triangles Are Similar

The two triangles below look like they could be similar but we cannot say for sure unless we know more about the length of the sides and/ or the angles within the triangle.

two triangles that appear to be similar

To find if triangles that are similar we must compare corresponding sides and/ or corresponding angles. The two examples below show corresponding sides and corresponding angles.

two similar triangles ABC and DEF

AB and DE are corresponding sides

BC and EF are corresponding sides

CA and FD are corresponding sides

BAC and ∠EDF are corresponding angles

ACB and ∠DFE are corresponding angles

CBA and ∠FED are corresponding angles

two similar triangles ABC and XYZ

AB and XY are corresponding sides

BC and YZ are corresponding sides

CA and ZX are corresponding sides

BAC and ∠YXZ are corresponding angles

ACB and ∠XZY are corresponding angles

CBA and ∠ZYX are corresponding angles

There are several combinations of conditions that show whether two triangles to be similar.

If all three pairs of corresponding sides are in the same ratio then the triangles are similar. If two of the corresponding angles are equal then the triangles are similar.
two similar triangles ABC and DEF

If,

AB = BC = CA
DE EF FD

then

ΔABC is similar to ΔDEF

 

 

If,

∠BAC = ∠EDF,

∠ACB = ∠DFE, and

∠CBA = ∠FED

then

ΔABC is similar to ΔDEF

Two points to note:

  1. If two pairs of corresponding angles are equal then the third pair will always be equal too (since the sum of the three angles in a triangle is always 180°).
  2. If one set of the conditions (e.g. corresponding sides in the same ratio) is true then the other set (e.g. corresponding angles being equal) is also true.

Another set of conditions that combine to show similarity in triangles is when two pairs of corresponding sides are in ratio and the of pair of corresponding angles included between these sides are equal. The two right-angled triangles below show an example of this.

two similar right triangles.

Similar Triangle Problems

Are the two triangles below similar?

two traingles that appear similar

We know the lengths of each side so if we can show that the corresponding sides are in proportion then the triangles are similar. So first we need to match the corresponding sides.

Match the hypotenuses (the longest sides)

two triangles with corresponding sides shown

Side of length 3.5 corresponds to side of length 5

Match the shortest sides

two triangles with corresponding sides shown

Side of length 1.4 corresponds to side of length 2

Match the remaining sides

two triangles with corresponding sides shown

Side of length 2.8 corresponds to side of length 4

Note: It does not matter which triangle you look at first when matching sides. Just keep the same order when writing the lengths for each pair of corresponding sides.

If the ratios of the lengths of the corresponding sides are equal then the triangles are similar.

AB = BC = CA
DE EF FD

3.5 ÷ 5 = 0.7

1.4 ÷ 2 = 0.7

2.8 ÷ 4 = 0.7

The ratios are equal so the triangles are similar.

The problem below is an example of how the properties of similar triangles can be used to solve real-life problems that can arise.

Joe is fed up worrying about whether his neighbors can see into his living room from their house. He has decided to build a fence that is high enough to block the view from their top floor window. He needs to work out how high to build the fence.

illustration of two adjacent houses

Joe has taken some measurements and has sketched them out as shown below.

sectional view of two adjacent houses

We can use what we know about similar triangles to find the height that Joe should make his new fence.

We can create a drawing based on Joe's sketch.

We will call the missing dimension that we need to find 'x'.

digram showing section between two houses

We have two similar triangles

two similar triangles with unknown side, x, shown

We can calculate the value of x.

1.8 = 8
x 3

8x = 3 x 1.8

8x = 5.4

x = 0.675m

Looking back at our drawing we can see that we need to add the value we calculated for x onto 2.0m to find the minimum height of the fence.

0.675m + 2.0m = 2.675m

Joe's fence must be at least 2.675m tall to block the neighbor's view. That is a fairly tall fence. Maybe he should get drapes!

Similarity Worksheets

Click the links below and get your child to try the worksheets on similarity in triangles and in other similar figures.

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