ABCD is a trapezoid, and AB is parallel to CE. If BD = 8 and DE = 7, then find the ratio of the area of triangle ADB to the area of triangle CDE.
+128474 Since ADCB is a trapezoid, then AD is also parallel to BC
And note that angle EBC = angle BDA
And because AB is parallel to CE, then angle BEC = angle DBA
So triangle BEC is similar to triangle DBA
So BE and DB are corresponding sides
And the ratio of the area of DBA to DEC = (BD/ BE)^2 = (8/15)^2 = 64/225
And triangles EDC and BEC share equal altitudes so their areas are to each other as their bases
ED = 7 BE = 15
So area EDC = (7/15)area of triangle BEC ⇒ EDC/ BEC = 7/15 = 105/ 225
And the ratio of the area of DBA to BEC = (8/15)^2 = 64/225 ⇒ DBA / BEC = 64/225
So
(DBA / BEC) 64 / 225 [ DBA ] 64
__________ = ________ = ______ = ____
(CDE / BEC) 105 / 225 [ CDE ] 105
