Use dilations and scale factors to establish similarity, set up proportions between corresponding sides of similar figures, and relate the scale factor to changes in perimeter and area.

What this topic is asking

The Geometry category requires you to work with similarity through dilations (the G-SRT standards). On the Grade 10 MCAS you use a scale factor to relate similar figures, set up proportions between corresponding sides, apply the angle-angle criterion for similar triangles, and understand how a scale factor changes perimeter and area. The crucial idea, that area scales by the square of the linear factor, is a frequent test point.

Dilations and scale factor

A dilation enlarges or shrinks a figure from a fixed center by a scale factor $k$:

• $k > 1$ enlarges the figure; $0 < k < 1$ shrinks it.
• Every length is multiplied by $k$, but every angle stays the same, so the shape is preserved.

A dilation is not a rigid motion (it changes size), which is why dilated figures are similar rather than congruent. On a coordinate grid, a dilation centered at the origin sends $(x, y) \to (kx, ky)$.

Similar figures and proportions

Two figures are similar when corresponding angles are equal and corresponding sides are in the same ratio. The constant ratio is the scale factor. This lets you find an unknown side by proportion.

For similar triangles with $AB = 5$ corresponding to $DE$, and another pair $BC = 8$ corresponding to $EF = 24$, the scale factor from the first to the second is $\frac{24}{8} = 3$, so $DE = 5 \times 3 = 15$. Setting up the proportion $\frac{AB}{DE} = \frac{BC}{EF}$ with matching corresponding sides is the reliable method; the only trap is pairing the wrong sides.

The angle-angle criterion

For triangles, you do not need all three pairs of sides or angles to prove similarity. The angle-angle (AA) criterion says that if two angles of one triangle equal two angles of another, the triangles are similar. The third angles must then match too, because angles in a triangle sum to $180^\circ$.

AA is why two triangles formed by a transversal cutting parallel lines, or by an altitude in a right triangle, are often similar: shared or equal angles do the work.

Scale factor, perimeter, and area

When a figure is scaled by a linear factor $k$, the different measures scale differently:

• Perimeter (a length) scales by $k$.
• Area scales by $k^2$, because area is length times length.
• Volume scales by $k^3$, because volume is length cubed.

So doubling every side ($k = 2$) doubles the perimeter but quadruples the area. This is one of the most common MCAS traps: students multiply the area by the linear factor instead of its square.

Try this

Q1. Similar triangles have a scale factor of 4. A side of 2.5 cm corresponds to what length in the larger triangle?

• Cue. $2.5 \times 4 = 10$ cm.

Q2. A shape's sides are tripled. By what factor does its area increase?

• Cue. $3^2 = 9$ times.