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9 grade maths> In a Triangle ABC, angle A = 90 0 , CA=AB...

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In a Triangle ABC, angle A = 90⁰, CA=AB,
D is a point on BC produced. Prove that AD²=AC²+BD X CD

Dipankar Ghosh , 10 Years ago

Grade 9

1 Answers

Sumit Majumdar

Last Activity: 9 Years ago

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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