Theorem

Similar triangles have proportional sides

Geometry • Triangle • Similarity

Formula

\frac{A ^{'} B ^{'}}{A B} = \frac{B ^{'} C ^{'}}{B C} = \frac{C ^{'} A ^{'}}{C A} = k, \frac{[ A ^{'} B ^{'} C ^{'} ]}{[ A B C ]} = k^{2}

If two triangles are similar (AA, SAS or SSS similarity), corresponding sides are in a constant ratio

k

. Areas are then in the ratio

k^{2}

, and all corresponding lengths (altitudes, medians, perimeters) in the ratio

k

How to use it

△ A B C \sim △ P QR

with

A B = 4

P Q = 6

. If

[A B C] = 8

, find

[P QR]

Step-by-step

1
$k = \frac{P Q}{A B} = \frac{6}{4} = \frac{3}{2}$
Why: Ratio of corresponding sides.
2
$[P QR] = k^{2} [A B C] = \frac{9}{4} \times 8 = 18$
Why: Areas scale as the square of the linear ratio.

Answer

18

Watch out

Areas scale as

k^{2}

and volumes as

k^{3}

— using

k

for area is the standard trap.

Builds on

k

Also known as

AA similarity, proportional sides, area ratio of similar figures