In trapezoid ABCD, the extensions of the lateral sides AB and CD meet at point F.

In trapezoid ABCD, the extensions of the lateral sides AB and CD meet at point F. a) Prove that triangle BFC and AFD are similar. b) Find the area of the trapezoid ABCD if AB: BF = 3: 1, and the area of the triangle BFC is 2cm2.

a) Consider triangles BFC and AFD: angle F is common, angle FBC = angle FAD (corresponding angles for parallel BC and AD and secant AB). This means that the triangles are similar (in two corners).

b) AB refers to BF as 3 to 1, let us denote AB as 3x and BF as x.

Let’s calculate the coefficient of similarity of triangles BFC and AFD:

k = AF / BF. AF = AB + BF = 3x + x = 4x. k = 4x / x = 4.

Let us express the area of ​​the triangle FBC: SFBC = 1/2 * BC * h.

Since the coefficient of similarity is 4, the side AD will be 4BC, and the height of the triangle AFD will be 4h.

Let us express the area of ​​the triangle AFD: SAFD = 1/2 * 4BC * 4h = 16 * (1/2 * BC * h).

That is, the area of ​​the triangle AFD is 16 times the area of ​​the triangle FBC, SAFD = 2 * 16 = 32 cm².

The area of ​​the trapezoid is equal to the difference between the areas of the triangles AFD and FBC.

SABCD = 32 – 2 = 30 cm².