What is similarity?
In math, two shapes are similar if they are the same shape and their corresponding sides are in proportion to each other. The ratio between the corresponding sides is called the scale factor. In addition, for two shapes to be similar their corresponding angles must be congruent.
How to use similarity:
Two shapes are similar if :
- They are the same shape. (They do not have to be the same size.)
- Their corresponding sides are proportionate to each other.
- Corresponding angles are congruent.
Scale factor:
The ratio between the corresponding sides of two similar shapes is called the scale factor.
To determine if two shapes are similar, first make sure they are the same shape. Remember they do not need to be the same size.
In the example below, the two triangles are the same shape and their corresponding angles are congruent. As one can see Angle B and Ange E correspond to each other and are both 105°.
To determine if these two triangles are similar, we must also look at whether or not the corresponding sides are in proportion to each other. To find this we look for the scale factor.
In this example, sides AB and ED are corresponding sides.
Side AB = 6 inches
Side ED = 2 inches
Side ED is ⅓ side EB.
1/3 is our scale factor.
To check if the triangles are similar, we must now check to see if the other corresponding sides have a scale factor of ⅓ as well.
Side BC and Side EF are corresponding sides.
Side BC = 9 inches
Side EF = 3 inches.
Side EF is ⅓ side BC.
Since the corresponding sides are all proportional to each other with a scale factor of ⅓, these two triangles are similar.
Sample Math Problems
1. Fill in the missing value.
Solution: For this problem, we need to look at the proportion. We see that XY is the corresponding side to QR, so we need to find what the corresponding side to YZ is. When we look at the two trapezoids, we see that RS is the corresponding side to YZ.
Therefore, the missing value is RS.
2. Find the missing number.
Solution: To solve this problem, we can create a proportion.
Now fill in the given information to get:
To solve this we get:
3.5(QR) = 2(1.75)
3.5(QR) = 3.5
Divide both sides by 3.5.
QR = 1 unit
3. Find the length of B’C’. All lengths are expressed in cm.
Solution: To solve this problem, we can create a proportion.
Now fill in the given information to get:
To solve this we get:
12(B’C’) = 8(9)
12(B’C’) = 72
Divide both sides by 12.
B’C’= 6 cm
4. Triangle ABC is similar to triangle DEF. What is the length of DE?
Solution: To solve this problem, we can create a proportion.
To solve this we get:
5(DE) = 12(24)
5(DE) = 288
Divide both sides by 5.
DE = 57.6 in.
