△ABC is a right triangle at B, with AB=3 and BC=4. We take point D on BC , such that BD=1. We construct the perpendicular at D meeting AC at E. Find the ratio of areas of the yellow region to that of the blue region. That is, find [ABE]/[BDE].
The ratio of sides is 3, 4, 5.
So BED sides are 1, 4/3, 5/3.
and BAE sides are 9/5, 12/5, 3.
The ratio between those sides are 9/5, so the area is just 81/25.
Our final answer is 81/25.
I hope this helped. :)))
Draw a parallel to BC from E intersecting AB at F
Then triangle AFE is similar to triangle ABC
And since AB = 3/4 of BC
Then AF =3/4 of EF = 3/4 (BD) = (3/4) (1) =3/4
Then AFE is a right triangle with legs AF and EF
So its area (1/2) (1) (3/4)= 3/8
And FEDB is a rectangle with a height of AB - AF = (3 - 3/4) = 9/4 and a width of 1
So its area = (1)(9/4) = 9/4
So the blue area and the yellow area inside this rectangle both equal 1/2 of this = 9/8
The yellow area = 3/8 +9/8 = 12/8
And the blue area = 9/8
And their ratio = (12/8) / (9/8) = 12/9 = 4/3