Question

In the figure, AB||CB. If CD:BA = 6:5 and the area of CED is 288, find the area of BEA

Answer 1

Since AB and CD are parallel, and they intersect, the smaller triangle is similar to the large. They will have the same ratio of sides and congruent angles. Thus, their bases are in a ratio and their heights should be in the same ratio, and their areas will be proportional.
5b1:6b2
5h1:6b2
Area of large triangle:
1/2 bh = 1/2 (6)6x = 288
18x = 288
x = 16 = ratio factor to be applied to smaller
So now the small triangle is:
1/2 (5)(5x) = (25/2)x = (25/2)(16) = 25×8 = 200

Answer 2

The area of BEA is 200 square units.

Explanation:

Given information: AB||CD and CD:BA = 6:5.

Let the length of sides CD and BA are 6x and 5x respectively.

If a transversal line intersect two parallel lines, then the alternative interior angles are equal.

`\angle EAB=\angle EDC` (Alternate interior angles)

`\angle EBA=\angle ECD` (Alternate interior angles)

`\angle AEB=\angle DEC` (Vertically opposite angles)

By AAA property of similarity,

`\triangle ABE\sim \triangle DEC`

The area of two similar triangles is proportional to the square of their corresponding sides.

`(Area(CED))/(Area(BEA))=(CD^2)/(BA^2)`

`(288)/(Area(BEA))=((6x)^2)/((5x)^2)`

`(288)/(Area(BEA))=(36x^2)/(25x^2)`

`(288)/(Area(BEA))=(36)/(25)`

`288* 25=36* Area(BEA)`

`7200=36* Area(BEA)`

Divide both sides by 36.

`200=Area(BEA)`

Therefore the area of BEA is 200 square units.