Geometry • Topic 4

Similarity

Two figures are similar ( $\sim$ ) if they have the same shape but not necessarily the same size. Similarity is essentially "scaling."

Similarity Criteria

Theorem (AA (Angle-Angle))

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. (Since angles sum to $18 0^{\circ}$ , this implies all three angles are equal).

Theorem (SAS Similarity)

If two sides are proportional (

\frac{A B}{D E} = \frac{A C}{D F}

) and the included angles are equal (

∠ A = ∠ D

), the triangles are similar.

Theorem (SSS Similarity)

If all three corresponding sides are proportional, the triangles are similar.

Properties

If $△ A BC \sim △ D EF$ with scale factor $k$ :

Corresponding Angles: Equal.
Corresponding Lengths: Ratio is $k$ (sides, medians, altitudes, perimeter).
Area Ratio: Ratio of areas is $k^{2}$ .

\frac{Area ( A BC )}{Area ( D EF )} = (\frac{A B}{D E})^{2}

Parallel Lines and Similarity

A line drawn parallel to one side of a triangle cuts off a triangle similar to the original. If

D E ∥ BC

△ A BC

, then

△ A D E \sim △ A BC

Practice Problems

Exercise (Problem 1)

In trapezoid

A BC D

(

A B ∥ C D

), diagonals intersect at

P

. If

A B = 6

and

C D = 10

, find the ratio

A P / PC

Exercise (Problem 2)

△ A BC

, a line parallel to

BC

intersects

A B

D

and

A C

E

. If the area of

△ A D E

is half the area of

△ A BC

, find the ratio

A D / A B

Exercise (Problem 3 (Right Triangle Mean))

In a right triangle, the altitude to the hypotenuse divides the triangle into two smaller triangles similar to the original and to each other. Use this to prove

h^{2} = x y

(where

h

is altitude,

x, y

are segments of hypotenuse).