Ratio of Areas of Similar Triangles
📐 What Are Similar Triangles?
Two triangles are similar if they have the same shape but not necessarily the same size. This means:
- ✅ All corresponding angles are equal
- ✅ All corresponding sides are in the same ratio (called the scale factor)
If $\triangle ABC \sim \triangle DEF$, and the ratio of their corresponding sides is $k$, then:
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$
⭐ The Key Theorem: Area Ratio
If two triangles are similar with a side ratio of $k$, then the ratio of their areas is $k^2$.
Formally, if $\triangle ABC \sim \triangle DEF$ with scale factor $k = \frac{AB}{DE}$, then:
$$\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{AB}{DE}\right)^2 = k^2$$