### Sample Problem

The area of the following trapezoid is 24 cm2. If 5 times its longer base is equal to 7 times its shorter base, then what is the difference between the area of ∆DOC and the area of∆ AOB?

cm2

#### Solution

We label the sections as shown. Since 7AB = 5CD, we have

.

Since the height of ABC and ACD are the same, the ratio of their areas is the ratio of their bases, so

.

Thus, the ratio of ABC to ABCD is

,

so its area is 24 × 5/12 = 10 cm2. ADC = ABCD-ABC, so ADC = 24 - 10 = 14 cm2. We see that ADC and BDC have the same area because they have the same base and height. Thus,

ADC-DOC = BDC-DOC,

so (3) = (4) . Therefore, we find the difference of DOC and AOB to be

(1) - (2) = [ (1) + (4) ]-[ (2) + (3) ] = 14 - 10 = 4 cm2.
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