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.
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.
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.
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
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. |
ΔABC is similar to ΔDEF
∠BAC = ∠EDF,
∠ACB = ∠DFE, and
∠CBA = ∠FED
ΔABC is similar to ΔDEF
Two points to note:
- 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°).
- 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.
Similar Triangle Problems
Are the two triangles below 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)
Side of length 3.5 corresponds to side of length 5
Match the shortest sides
Side of length 1.4 corresponds to side of length 2
Match the remaining sides
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.
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.
Joe has taken some measurements and has sketched them out as shown below.
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'.
|We have two similar triangles |
We can calculate the value of x.
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!