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In a Triangle ABC, angle A = 90 0 , CA=AB, D is a point on BC produced. Prove that AD 2 =AC 2 +BD X CD

In a  Triangle ABC, angle A = 90, CA=AB,
D is a point on BC produced. Prove that AD2=AC2+BD X CD

Grade:9

1 Answers

Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
5 years ago
68-2201_1.jpg
From the figure, we see that:
AD^2=AE^2+ED^2=\left(AC+CE \right )^2+ED^2=AC^2+CE^2+AC\cdot CE+ED^2
From symmetry of triangles, we know that
AC\cdot CE=BC\cdot CD
So, we get
AD^2=AC^2+CE^2+BC\cdot CD+ED^2=AC^2+BC\cdot CD+CD^2=AC^2+BC\left(BC+CD \right )=AC^2+BD\cdot CD

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