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# area problem

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△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]. Jan 27, 2021

#1
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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.

I hope this helped. :)))

=^._.^=

Jan 27, 2021
#3
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It seems like I may have made a mistake on this answer. CPhill has the correct one. :)))

I'm sorry.

=^._.^=

catmg  Jan 27, 2021
#2
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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   Jan 27, 2021