SOLUTION: In triangle ABC ,angle ACB is 90 degrees and CD is perpendicular to AB.Prove CB<sup>2</sup>/CA<sup>2</sup>=BD/AD

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Question 174528: In triangle ABC ,angle ACB is 90 degrees and CD is perpendicular to AB.Prove CB2/CA2=BD/AD
Answer by jojo14344(1512) About Me  (Show Source):
You can put this solution on YOUR website!

In triangle ABC ,angle ACB is 90 degrees and CD is perpendicular to AB.
.
Let's draw the Triangle according to the condition:
drawing%28400%2C400%2C-6%2C1%2C-1%2C6%2Cgraph%28400%2C400%2C-6%2C1%2C-1%2C6%29%2Ctriangle%280%2C0%2C-5%2C0%2C0%2C5%29%2Cblue%28line%28-2.5%2C2.5%2C0%2C0%29%29%2Cblue%28locate%28-5.2%2C0.3%2CA%29%29%2Cblue%28locate%280.2%2C5.2%2CB%29%29%2Cblue%28locate%280.2%2C0.3%2CC%29%29%2Cgreen%28line%28-.5%2C0%2C-.5%2C.5%29%29%2Cgreen%28line%28-.5%2C.5%2C0%2C.5%29%29%2Cblue%28locate%28-2.7%2C2.8%2CD%29%29%2Cred%28locate%28-4.7%2C.5%2C45%5Eo%29%29%2Cred%28locate%28-2.7%2C2.2%2C90%5Eo%29%29%2Cred%28locate%28-1%2C.5%2C45%5Eo%29%29%2Cred%28locate%28-.5%2C1%2C45%5Eo%29%29%2Cred%28locate%28-.5%2C4.5%2C45%5Eo%29%29%2Cred%28locate%28-2.2%2C2.8%2C90%5Eo%29%29%29 -------> "CD" perpendicular to "AB"; Angle "ACB=90%5Eo"
.
In Trigonometric Function for Right Triangles, tan%2845%5Eo%29=1, and we know tangent=opp%2Fadjacent
And it shows in our Triangle, tan%2845%5Eo%29=highlight%28AD%29%2FCD=highlight%281%29%2F1=1
Also, tan%2845%5Eo%29=highlight%28BD%29%2FCD=highlight%281%29%2F1=1
.
By Pyth. Theorem: ----> hyp%5E2=opp%5E2%2Badj%5E2
CA%5E2=AD%5E2%2BCD%5E2=1%5E2%2B1%5E2=1%2B1
highlight%28CA%5E2=2%29
Also,CB%5E2=CD%5E2%2BBD%5E2=1%5E2%2B1%5E2=1%2B1
highlight%28CB%5E2=2%29
Therefore,
Prove CB%5E2%2FCA%5E2=BD%2FAD, substitute highlighted values:
2%2F2=1%2F1
1=1
Thank you,
Jojo