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Question

If p1,p2,p3 are respectively the perpendiculars from the vertices of a triangle to the opposite side, then p1p2p3 is equal to


A

a2b2c2

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B

2a2b2c2

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C

4a2b2c3R2

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D

a2b2c28R3

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Solution

The correct option is D

a2b2c28R3


Explanation for the correct option:

Step 1: Draw a required digram

Where, a,bandc are the sides of the triangle ABC.

We know that the area of the triangle =12×base×height

Step 2: Calculate the value of p1p2p3

Area of the triangle ABC

A=12×BC×ADA=12×a×p1p1=2Aa

Similarly,

A=12×AC×BEA=12×b×p2p2=2Ab

Similarly,

A=12×AC×CFA=12×c×p3p3=2Ac

Now,

p1p2p3=2Aa×2Ab×2Acp1p2p3=8A3abc

Step 3: Apply the concept A=abc4R (from sine rule of the triangle) where Ris the radius of the circumcircle.
Therefore, putting the value of A in the above expression,

p1p2p3=8(abc4R)3abc=8a3b3c364R3abc=a2b2c28R3

Hence, the correct option is (D).


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