Skill: Identify similar figures

Figures are similar if they have the same shape. To check, look for:

1. Matching corresponding vertices and angles.
2. A single scale factor that changes every side length by the same amount.
3. In a coordinate plane, a dilation may be combined with a translation, rotation, or reflection.

A scale factor greater than $1$ enlarges a figure. A scale factor between $0$ and $1$ reduces it. A scale factor of $1$ means the figures are congruent.

Example

Are $\triangle ABC$ and $\triangle DEF$ similar?

$$A(1,1),\quad B(3,1),\quad C(1,2)$$ $$D(4,2),\quad E(8,2),\quad F(4,4)$$

Step 1: Match the corresponding vertices

The horizontal and vertical positions show this matching:

• $A$ corresponds to $D$
• $B$ corresponds to $E$
• $C$ corresponds to $F$

Step 2: Compare corresponding side lengths

For $\triangle ABC$:

• $AB=2$
• $AC=1$
• $BC=\sqrt{(3-1)^2+(1-2)^2}=\sqrt{5}$

For $\triangle DEF$:

• $DE=4$
• $DF=2$
• $EF=\sqrt{(8-4)^2+(2-4)^2}=\sqrt{20}=2\sqrt{5}$

Now compare each pair:

$$\frac{DE}{AB}=\frac{4}{2}=2$$ $$\frac{DF}{AC}=\frac{2}{1}=2$$ $$\frac{EF}{BC}=\frac{2\sqrt{5}}{\sqrt{5}}=2$$

All corresponding sides have the same scale factor, $2$.

Step 3: Check the coordinate relationship

Multiplying the coordinates of $\triangle ABC$ by $2$ gives:

$$(1,1)\to(2,2),\quad (3,1)\to(6,2),\quad (1,2)\to(2,4)$$

Moving each point $2$ units to the right gives:

$$(2,2)\to(4,2),\quad (6,2)\to(8,2),\quad (2,4)\to(4,4)$$

These are exactly $D$, $E$, and $F$.

Therefore, $\triangle ABC$ and $\triangle DEF$ are similar. The second triangle is an enlargement of the first by a scale factor of $2$, followed by a translation.