What Are Similar Shapes?
Similar shapes have the same form but differ in size. This means their corresponding angles are equal, and their corresponding sides are in proportion. For example, two triangles are similar if their angles match exactly, even if one is larger than the other. Similarity preserves shape but not necessarily size.
- Corresponding angles are equal.
- Corresponding sides are proportional.
Identifying Corresponding Sides and Angles
To work with similar shapes, you must correctly identify which sides and angles correspond. Corresponding angles are those in the same relative position in each shape. Corresponding sides are the sides between those angles. For example, if angle A in one triangle corresponds to angle A in another, then the side opposite angle A in the first triangle corresponds to the side opposite angle A in the second triangle.
- Match angles by their position and measure.
- Identify sides between corresponding angles as corresponding sides.
Using Scale Factors to Compare Lengths
The scale factor is the ratio of any pair of corresponding lengths in two similar shapes. It tells you how much one shape has been enlarged or reduced compared to the other. For example, if one side in shape A is 3 cm and the corresponding side in shape B is 9 cm, the scale factor from A to B is 9 ÷ 3 = 3. This means every length in shape B is three times the length of the corresponding side in shape A.
- Scale factor = length in larger shape ÷ length in smaller shape.
- All corresponding lengths share the same scale factor.
Calculating Unknown Lengths Using Proportions
When some lengths are unknown, set up a proportion using the scale factor. For example, if two similar rectangles have a known side of 5 cm in the smaller and an unknown corresponding side x in the larger, and the scale factor is 2, then:
\[ \frac{x}{5} = 2 \implies x = 2 \times 5 = 10 \text{ cm} \]
This method works for any pair of corresponding sides.
- Set up proportion: (length in shape 1) ÷ (length in shape 2) = scale factor.
- Solve for the unknown length by cross-multiplying.
Worked Example
Two similar triangles have corresponding sides of lengths 6 cm and 9 cm. Find the length of a third side in the smaller triangle if the corresponding side in the larger triangle is 15 cm.
Step 1: Find the scale factor from smaller to larger triangle:
\[ \text{Scale factor} = \frac{9}{6} = 1.5 \] (since 9 cm corresponds to 6 cm) Step 2: Use the scale factor to find the unknown length x in the smaller triangle: \[ \frac{15}{x} = 1.5 \implies x = \frac{15}{1.5} = 10 \text{ cm} \] Step 3: Check that the scale factor is consistent: \[ \frac{15}{10} = 1.5 \] The calculation is correct.
- Calculate scale factor from known sides.
- Set up proportion with unknown length.
- Solve and check consistency.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify Corresponding Parts | Match angles and sides in both shapes | Angle A corresponds to angle A; side opposite angle A corresponds | Ensures correct sides are compared for length calculations |
| Calculate Scale Factor | Divide length in one shape by corresponding length in the other | Scale factor = 8 cm ÷ 4 cm = 2 | Use to find unknown lengths by multiplying or dividing |
| Set Up Proportion | Write ratio equation using scale factor | x ÷ 5 cm = 2 | Solve for x to find missing length |
| Solve and Check | Cross-multiply and calculate unknown length | x = 2 × 5 cm = 10 cm | Confirm scale factor consistency to avoid errors |
Active recall
Quick check
1. If two similar shapes have a scale factor of 3 from the smaller to the larger, and a side in the smaller shape is 7 cm, what is the length of the corresponding side in the larger shape?
Multiply the smaller length by the scale factor: 7 cm × 3 = 21 cm.
2. Two similar triangles have sides 5 cm and x cm in the smaller triangle, and 15 cm and 9 cm in the larger triangle. Find x.
Calculate scale factor from 5 cm to 15 cm: 15 ÷ 5 = 3. Use this to find x: 9 ÷ x = 3 → x = 9 ÷ 3 = 3 cm.
3. Are the corresponding angles in similar shapes equal or proportional?
Corresponding angles in similar shapes are equal.
4. If the scale factor from shape A to shape B is 0.5, how do lengths in shape B compare to shape A?
Lengths in shape B are half the lengths of the corresponding sides in shape A.