△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].

Guest Jan 27, 2021

#1**0 **

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. :)))

=^._.^=

catmg Jan 27, 2021

#2**+1 **

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

So

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

CPhill Jan 27, 2021