Question #aa5c1

1 Answer
Mar 3, 2017

#144# area units.

Explanation:

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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#