Identify similar and congruent figures, use proportional corresponding sides and equal corresponding angles, and apply scale factors to lengths, areas and volumes (Geometry).

What this topic is asking

Similarity and congruence describe figures with the same shape. The ACT tests recognising similar figures (equal angles, proportional sides), solving for an unknown side with a proportion, and using a scale factor, including how it affects area and volume. The key fact, often tested, is that area scales by the square and volume by the cube of the length scale factor.

Similar versus congruent

The distinction is size.

Triangles are similar if two pairs of angles match (AA), and congruent under criteria such as SSS, SAS or ASA.

Solving with proportions

When two figures are similar, corresponding sides form equal ratios.

Aligning corresponding sides correctly (matching the small triangle's side to the large triangle's matching side) is the crucial step.

Scale factor and area and volume

This is the most-tested subtlety. If similar figures have a length scale factor $k$:

• Lengths (sides, perimeters) scale by $k$.
• Areas (surface area too) scale by $k^{2}$.
• Volumes scale by $k^{3}$.

So doubling every length ($k = 2$) multiplies area by $4$ and volume by $8$. A model built at $\frac{1}{10}$ scale has $\frac{1}{100}$ the surface area and $\frac{1}{1000}$ the volume. Forgetting to square or cube the factor is the classic mistake.

Recognising similar triangles in a figure

Many ACT problems hide similar triangles inside a single figure. A line parallel to one side of a triangle cuts the other two sides proportionally and creates a smaller triangle similar to the whole. A right triangle with an altitude drawn to its hypotenuse splits into two smaller triangles, each similar to the original and to each other. And two triangles sharing an angle, with a parallel cut, are similar by AA. Spotting these configurations is what lets you set up a proportion; once you see which triangles are similar, the matching sides give the equation directly.

Indirect measurement with similar triangles

A classic application is indirect measurement: using shadows or sight lines to find a height you cannot reach. If a person 6 feet tall casts a 4-foot shadow at the same time a tree casts a 20-foot shadow, the person and the tree (with their shadows) form similar right triangles, so $\frac{6}{4} = \frac{h}{20}$, giving $h = 30$ feet. The Sun's rays make equal angles, which is why the triangles are similar. Setting up the proportion of corresponding sides, height over shadow for each object, solves the problem in one step.

Why proportional reasoning is the core skill

Every similarity question reduces to a proportion or a scale factor. Set up corresponding sides as equal ratios, solve, and, when area or volume is involved, raise the length factor to the second or third power. Recognising similar triangles by equal angles (often from parallel lines or shared angles in a figure) is what lets you write the proportion in the first place. Keeping corresponding parts aligned and powering the scale factor correctly secures these points.

Try this

Q1. Two similar triangles have sides in ratio 3 to 4. A side of the smaller is 9. Find the matching side of the larger. [1 point]

• Cue. $\frac{3}{4} = \frac{9}{x}$, so $x = 12$.

Q2. Two similar solids have length scale factor 2. How many times larger is the volume of the bigger one? [1 point]

• Cue. Volume scales by $k^{3} = 2^{3} = 8$.