Similar Triangles
Lesson · Beginner
Geometry: Plane Geometry

If two triangles are similar with side ratio
then their areas have ratio
In trapezoid ABCD, AB ∥ CD and diagonals AC and BD intersect at P.
Suppose [ABP] = 25 and [CDP] = 64
Find the area of trapezoid ABCD.
Because AB ∥ CD, we have
and

Therefore,
Since their areas are 25 and 64, their corresponding side ratio is

Now triangles ABP and BCP share the same altitude from B to AC, so
From similarity,
Thus
Similarly,
Therefore,
In triangle ABC, point D lies on BC with BD : DC = 2 : 3
Through D, draw DE ∥ AC and DF ∥ AB, where E lies on AB and F lies on AC.
If [ABC] = 150, find the area of quadrilateral AEDF.

Since DE ∥ AC, we have
with scale factor
Therefore,
so
Similarly,
with scale factor
Thus
Hence
Two lines parallel to BC divide triangle ABC into three regions of equal area.

Find AD : DE : EB

By (1), (3)
By (2), (4)