Question

What is the ratio between area of rectangle abcd and triangle bec?

Answer 1

-A2A-

Let M and N be the foot of perpendicular from F on BE and CD respectively.

Let AE = x , AD = BC = y and FM = h.

So, BE = 3*AE = 3x

FN = y - FM = y - h

Area of rectangle ABCD, Ar.(ABCD) =(4x)∗y =4xy

In △s BFE and CFD,

∠FBE =∠FDC (Alternate Interior angles)

∠BFE =∠CFD (Vertically Opposite angles)

By AA similarity, △BFE∼△CFD

In similar triangles, corresponding sides are proportional.

⇒BEFM =CDFN

3xh =4xy−h

h=3y7

So, area of triangle BFE = 12∗BE∗FM =914xy

Area of triangle BEC = 12∗BE∗BC =32xy

Area of triangle BFC = Area of triangle BEC - Area of triangle BFE

Ar.(△BFC) = 32xy−914xy

=67xy

Ratio of area of rectangle ABCD to the area of triangle BFC = Ar.(△BFC)Ar.(ABCD)

= 67xy4xy

= 314