Similarity

Two shapes are similar if one can be obtained from the other by a UNIFORM SCALING (and optionally a rotation/reflection/translation). The two shapes have:

• THE SAME ANGLES at corresponding vertices.
• PROPORTIONAL CORRESPONDING SIDES.

The scale factor $k$ is the ratio of corresponding sides: $k = \frac{\text{side of larger shape}}{\text{side of smaller shape}}.$

If $k > 1$: enlargement. If $k < 1$: reduction. If $k = 1$: shapes are CONGRUENT (identical).

Worked example. Two triangles. Smaller has sides $3, 4, 5$. Larger has sides $6, 8, 10$. Are they similar?

• Compare: $6/3 = 2$, $8/4 = 2$, $10/5 = 2$. All sides in the same ratio (2).
• Yes, similar with scale factor $k = 2$.

Three tests for similarity of triangles:

1. AAA (angle-angle-angle): all three angles equal. (Since angles sum to $180°$, two equal is enough — AA suffices.)
2. SSS (side-side-side in ratio): all three sides proportional.
3. SAS (side-angle-side): two sides proportional with the INCLUDED angle equal.

When you SHOW two triangles are similar, state which test you're using.

Once you know two shapes are similar, ratios of corresponding sides are equal.

Worked example. Two similar triangles. The smaller has sides $4, 6, 8$. The corresponding side to '4' in the larger is '10'. Find the other sides of the larger triangle.

• Scale factor: $k = 10/4 = 2.5$.
• Other sides: $6 \times 2.5 = 15$ and $8 \times 2.5 = 20$.
• Larger triangle has sides $10, 15, 20$.

Worked example (in proof form). Triangle $ABC$ with $\angle A = 90°$. A line drawn parallel to $BC$ cuts $AB$ at $D$ and $AC$ at $E$. Show that $\triangle ADE \sim \triangle ABC$.

• $\angle A$ is common.
• $DE \parallel BC$, so corresponding angles: $\angle ADE = \angle ABC$ and $\angle AED = \angle ACB$.
• All three angles match → AAA → $\triangle ADE \sim \triangle ABC$.

This is a CLASSIC criterion-C justification problem. Spell out which angles are equal and why.

If two similar shapes have linear scale factor $k$ (any corresponding length ratio):

• AREA scale factor = $k^2$.
• VOLUME scale factor = $k^3$.

Why? Area is length × length, so scaling both by $k$ scales area by $k^2$. Volume is length³, so scales by $k^3$.

Linear	Area	Volume
2	4	8
3	9	27
0.5	0.25	0.125
10	100	1000

Worked example. Two similar cones. The smaller has volume $50\,\text{cm}^3$. The radii are in the ratio 2:5. Find the volume of the larger cone.

• Linear scale factor: $k = 5/2 = 2.5$.
• Volume scale factor: $k^3 = 15.625$.
• Larger volume: $50 \times 15.625 = 781.25\,\text{cm}^3$.

Worked example. Two similar rectangles. Their areas are $20\,\text{cm}^2$ and $45\,\text{cm}^2$. Find the linear scale factor between them.

• $k^2 = 45/20 = 2.25$.
• $k = \sqrt{2.25} = 1.5$.

This is a HUGE Extended-level topic. Watch out for the difference between linear, area, and volume scale factors.