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Question

If r1,r2,r3 be the radii of excircles of the triangle ABC then r1r1r2 is equal to.


A

cotA2

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B

cotA2cotB2

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C

tanA2

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D

tanA2

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Solution

The correct option is C

tanA2


Explanation for the correct option:

Step 1: Concept and formula to be used.

According to the Herons formula, Radii of excircles of a triangle can be given by

r1=s-ar2=s-br3=s-c

Where, is the area of the triangle given by =ss-as-bs-c.

r1,r2,r3 are the radii of excircles.

a,b,c are the length of the sides of the triangle.

s is the semi-perimeter of the triangle given by s=12a+b+c.

tanA2=s-bs-css-atanB2=s-as-css-btanC2=s-as-bss-c

Step 2: Compute the required value.

Expand the given expression as follows:

r1r1r2=r1+r2+r3r1r2+r2r3+r1r3

Substitute the values of radii of excircles.

r1r1r2=s-a+s-b+s-cs-as-b+s-bs-c+s-cs-ar1r1r2=s-a+s-b+s-c2s-as-b+2s-bs-c+2s-cs-ar1r1r2=s-a+s-b+s-c1s-as-b+1s-bs-c+1s-cs-ar1r1r2=1s-a+1s-b+1s-c1s-as-b+1s-bs-c+1s-cs-ar1r1r2=1s-a+1s-b+1s-cs-a+s-b+s-cs-as-bs-cr1r1r2=1s-a+1s-b+1s-c3s-a+b+cs-as-bs-cr1r1r2=1s-a+1s-b+1s-c3s-2ss-as-bs-cr1r1r2=1s-a+1s-b+1s-css-as-bs-c

Multiply and divide by s inside the square root.

r1r1r2=1s-a+1s-b+1s-cs2ss-as-bs-cr1r1r2=1s-a+1s-b+1s-csr1r1r2=ss-a+ss-b+ss-c [Since, =ss-as-bs-c]

]Which can be written as follows:

r1r1r2=s-bs-css-a+s-as-css-b+s-as-bss-cr1r1r2=tanA2+tanB2+tanC2r1r1r2=tanA2

Hence, option C is the correct answer.


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