Find that area

Geometry Level 4

A B C ABC is a triangle. D D is a on point A B AB such that A D AD = 20 20 , B D BD = 30 30 . E E and F F are points on B C BC and C A CA respectively such that [ A B E ] = [ D B E F ] [ABE]=[DBEF] . Find [ A B E ] [ABE] if [ A B C ] = 75 [ABC]=75 .
Note: [ P Q R S ] [PQRS] denotes the area of the figure P Q R S PQRS .


The answer is 45.

Mietantei Conan by mietantei conan , 24, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Mietantei Conan
May 26, 2014

Suppose A E AE meets D F DF at P P Join the points D D and E E . It is given that
[ A B E ] = [ D B E F ] [ABE]=[DBEF] and this implies
[ A D P ] = [ P F E ] [ADP]=[PFE] (as common areas will get cancelled). Triangles A D E ADE and F D E FDE have same areas and same base; this implies A F ∣ ∣ D E AF||DE or A C ∣ ∣ D E AC||DE . So triangles D B E DBE and A B C ABC are similar and from this we get B E / B C = B D / A B = 3 / 5 BE/BC=BD/AB=3/5 . Therefore,
[ A B E ] = [ A B C ] × 3 / 5 = 75 × 3 / 5 = 45 [ABE]=[ABC]×3/5=75×3/5=45


2 Helpful 0 Interesting 0 Brilliant 0 Confused

Great problem...good solution

Sanjana Nedunchezian - 6 years, 9 months ago

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...