Similar figures and similar triangles

Two figures are similar if one is a scaled copy of the other , same shape, possibly different size. Two squares are always similar (any two squares!). Two circles are always similar. Two rectangles are similar only when their sides are proportional.

Definitions

Two polygons $P_1$ and $P_2$ are similar if:

1. their corresponding angles are equal, and
2. their corresponding sides are in the same ratio.

For triangles, this simplifies considerably (as we will see). For now:

Two triangles $\triangle ABC$ and $\triangle DEF$ are similar (written $\triangle ABC \sim \triangle DEF$) if $\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F,$ and $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}.$

The notation $\triangle ABC \sim \triangle DEF$ encodes the correspondence: $A \leftrightarrow D, B \leftrightarrow E, C \leftrightarrow F$. So $AB$ corresponds to $DE$, etc. Always write the letters in the matching order.

Key facts

A few cleanly stated facts:

• All congruent figures are similar, but not vice versa. Congruent = similar with ratio $1$.
• All circles are similar. (Same shape, possibly different radius.)
• All squares are similar. (Equal angles automatic, sides proportional automatic.)
• Two rectangles are similar iff their side ratios are equal. $2 \times 4$ and $3 \times 6$ are similar; $2 \times 4$ and $3 \times 5$ are not.
• Two equilateral triangles are always similar. (Same angles, same shape.)

For triangles specifically, we will prove that just the "equal angles" condition forces the "proportional sides" condition (the AAA criterion). So for triangles, similarity is much easier to verify than for general polygons.

Why similarity matters

Similar figures preserve all angles and ratios of corresponding lengths. So if $\triangle ABC \sim \triangle DEF$ with ratio $k$ (meaning $DE = k \cdot AB$ etc.), then:

• Every length in $\triangle DEF$ is $k$ times the corresponding length in $\triangle ABC$.
• The perimeter of $\triangle DEF$ is $k$ times the perimeter of $\triangle ABC$.
• The area is $k^2$ times , we prove this in topic 4.

So similar figures form a beautiful, scalable family, and they let us do indirect measurement: we can find the height of a tall tree by measuring its shadow and a similar shadow of a stick of known height.

Worked examples

Example 1. $\triangle ABC$ has $\angle A = 60^\circ, \angle B = 80^\circ$. $\triangle DEF$ has $\angle E = 80^\circ, \angle F = 40^\circ$. Are they similar? If so, write the correspondence.

$\angle C = 180^\circ - 60^\circ - 80^\circ = 40^\circ$. $\angle D = 180^\circ - 80^\circ - 40^\circ = 60^\circ$. So $\angle A = \angle D = 60^\circ$, $\angle B = \angle E = 80^\circ$, $\angle C = \angle F = 40^\circ$. Triangles are similar: $\triangle ABC \sim \triangle DEF$.

Example 2. Two triangles have sides $3, 4, 5$ and $6, 8, 10$. Are they similar?

All ratios equal $1/2$. Yes, similar.

Example 3. Triangles $\triangle PQR \sim \triangle XYZ$ with $PQ = 6, QR = 8, RP = 10$ and $XY = 9$. Find $YZ$ and $ZX$.

Ratio $PQ/XY = 6/9 = 2/3$. So $YZ = QR \cdot 3/2 = 12$ and $ZX = RP \cdot 3/2 = 15$.

Example 4. Are all isoceles triangles similar? Justify.

No. Two isoceles triangles can have different apex angles. E.g., $50^\circ, 65^\circ, 65^\circ$ versus $80^\circ, 50^\circ, 50^\circ$ , both isoceles but not similar.

Example 5. The perimeters of two similar triangles are $30$ cm and $45$ cm. If one side of the first triangle is $12$ cm, find the corresponding side of the second.

Ratio of perimeters $=$ ratio of sides $= 30/45 = 2/3$. Corresponding side $= 12 \cdot 3/2 = 18$ cm.

Try it yourself

1. Are all squares similar? All rectangles similar? Justify.
2. Two right triangles, both with a $30^\circ$ angle, are similar. Why?
3. $\triangle ABC$ has $\angle A = 70^\circ, \angle B = 50^\circ$. $\triangle PQR$ has $\angle P = 70^\circ, \angle Q = 60^\circ$. Are they similar?
4. Sides $4, 7, 8$ and $12, 21, 24$. Similar?
5. $\triangle ABC \sim \triangle DEF$ with $AB = 6, BC = 9, CA = 12, DE = 4$. Find $EF$ and $FD$.
6. Two equilateral triangles have sides $a$ and $b$. The ratio of perimeters? Of areas?
7. Sketch two similar pentagons of different sizes.
8. If $\triangle ABC \sim \triangle DEF$ and $\triangle DEF \sim \triangle GHI$, must $\triangle ABC \sim \triangle GHI$? (Transitivity.)
9. A photo is enlarged by a factor of $4$. By what factor does its area change?
10. State one real-life situation where similar triangles are used.

Pitfalls / Insight

• The correspondence matters. $\triangle ABC \sim \triangle DEF$ is different from $\triangle ABC \sim \triangle EDF$.
• Congruent $\Rightarrow$ similar, but not vice versa.
• Equal angles is automatic for some shapes (squares, equilateral triangles, regular $n$-gons) , proportional sides is then the only condition.

Insight. Similar figures are nature's photocopiers. Once you spot a "scaled copy", almost any unknown length or area can be recovered by ratio.