Question

Question #aa5c1

Answer 1

`144` area units.

Explanation:

Let

`S= abs(Delta ABC)`
`S_1=abs(Delta PGH)`
`S_2=abs(Delta PID)`
`S_3=abs(Delta PFE)`

by similarity of triangles we have

`S_1/S = (bar(HP)/bar(AB))^2`
`S_2/S = (bar(ID)/bar(AB))^2`
`S_3/S = (bar(PE)/bar(AB))^2`

because in two similar triangles, the ratio of their areas is equal to the square of the ratio of their respective sides.

But `bar(HP)=bar(AI)` and `bar(PE)=bar(DB)`

so

`sqrt(S_1/S)+sqrt(S_2/S)+sqrt(S_3/S)=(bar(AI)/bar(AB))+(bar(ID)/bar(AB))+(bar(DB)/bar(AB))=1`

then

`S=(sqrt(S_1)+sqrt(S_2)+sqrt(S_3))^2`

or

`S = (3+4+5)^2=144`