Similar shapes & missing lengths

Why equal angles force sides into a fixed ratio, finding the scale factor the right way round, similar triangles hidden inside one figure, and the area and volume scale factors k² and k³.

Geometry & measures · Congruence & similarity

Similar shapes & missing lengths

Why equal angles force sides into a fixed ratio, finding the scale factor the right way round, similar triangles hidden inside one figure, and the area and volume scale factors k² and k³.

Why it works

Similar means same shape, different size. Corresponding angles are equal and corresponding sides are all in the same ratio — that ratio is the scale factor

k

. For triangles, equal angles are enough: if two triangles have the same three angles, their sides must be in proportion.

Finding the scale factor: image over object. Pick a pair of corresponding sides where you know both lengths and divide the one in the shape you're going to by the one you're coming from:

k = \frac{\text{length on the new shape}}{\text{matching length on the old shape}}.

Then every other length on the new shape is

k \times

its partner. Getting this upside down is the single commonest error — sense-check it: if the new shape is bigger,

k > 1

.

Match the sides correctly. Corresponding sides are the ones *opposite equal angles*, not the ones in the same position on the page. When a question gives similar triangles in different orientations, redraw them the same way up, or list the vertices in matching order and pair the sides off from that list.

Triangles hidden inside one figure. A line drawn parallel to one side of a triangle cuts off a smaller triangle similar to the whole one (the parallel makes corresponding angles equal). Two more classics: the two triangles formed by crossing lines between parallel ends ("bow-tie" or "hourglass"), and the pair formed when a perpendicular drops from the right angle of a right-angled triangle. In every case, the trap is using the part where the whole is needed: in the wedge picture, the big triangle's side is the small one plus the extra piece, not the extra piece alone.

Area and volume scale differently — and this is where marks are won. If lengths scale by

k

\text{areas scale by } k^2, \qquad \text{volumes scale by } k^3.

An area is two lengths multiplied, a volume three. So doubling every length gives

4\times

the area and

8\times

the volume. Working backwards, a known area ratio gives

k = \sqrt{\text{area ratio}}

and a known volume ratio gives

k = \sqrt[3]{\text{volume ratio}}

— take the root first, then use

k

on the lengths.