In the parallelogram ABCD above, AD = 12. If the area of triangle ABE

If we take AD as the base of both the parallelogram and of the triangle ADE, and take EC as the base of triangle ECD, notice that all three have the same height - the perpendicular (vertical) distance between the two sides of the parallelogram.

If h is that height, and b is the length of AD, then the area of the parallelogram is bh, and the area of triangle ADE is bh/2. So triangle ADE takes up half the area of the parallelogram, and we know ABE takes up another 3/8 of the area, which means triangle ECD takes up the rest, or 1/8 of the entire area. So the area of ADE is 4 times the area of ECD (one is 1/2 the parallelogram's area, the other 1/8), and since they have the same height, the base of ADE must be 4 times the base of ECD. Since the base of ADE is 12, the base of ECD must be 3, so that's the length of EC.
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BE: 12-x
(12-x)*h/2=3/8*12*h
(12-x)*8=3*2*12
96-8x=72
x=3

EC=x=3

Area of triangle ABE = 1/2*BE*AG
Area of parallelogram ABCD = AD* BF
Given 1/2*BE*BF=3/8*AD*BF
But AG=BF and AD=12
=> 1/2*BE*BF=3/8*AD*BF
=> BE=9
CE=12-9=3

We can let the height of the parallelogram be 10. Notice that this will the be height of the three triangles we see in the diagram.

Therefore, the parallelogram has an area of 12 x 10 = 120 and hence the area of triangle ABC = 3/8 x 120 = 45. Triangle AED is half of the area of the parallelogram; thus, it has an area 1/2 x 120 = 60. Therefore, the area of triangle EDC is 120 - 45 - 60 = 15, and we can create the following equation to find EC:

1/2 x EC x 10 = 15

5 x EC = 15

EC = 3
Alternate Solution:

Let the height of the parallelogram be h. Then, the area of the parallelogram is 12h.

Notice that the height of triangle ABE is also h and, in terms of |BE| and h, the area of the triangle ABE is (|BE|*h)/2. We know that this is equal to 3/8 of 12h, so let’s set up the following equation:

(|BE|*h)/2 = (3/8)*12h

|BE|*h = (3/4)*12h = 3 * 3h = 9h

|BE| = 9

Since |BE| = 9, |EC| = |BC| - |BE| = |AD| - |BE| = 12 - 9 = 3.

Answer: D
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ABCD is para
ABE is triangle 3/8 para
AECD is a trapezium 5/8 para

Area para = bh = 12h
Area trapezium = b1+b2/2*h… 12+EC/2*h = 5/8*12h
…12+EC/2=5/8*12…12+EC=60/8*2…
EC=60/4-12=15-12=3

Ans (D)

Height of triangle = height of parallelogram = H

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