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Maths-

General

Easy

Question

In triangle $A B C comma blank$ if $tan invisible function application open parentheses A divided by 2 close parentheses equals 5 divided by 6$ and $tan invisible function application open parentheses B divided by 2 close parentheses equals 20 divided by 37 comma blank$ the sides $a comma blank b blank$ and $blank c$ are in

A.P.
GP.
H.P
None of these

The correct answer is: A.P.

We have
$tan invisible function application fraction numerator C over denominator 2 end fraction equals tan invisible function application open parentheses 90 degree minus fraction numerator A plus B over denominator 2 end fraction close parentheses$
$equals cot invisible function application fraction numerator A plus B over denominator 2 end fraction equals fraction numerator cot invisible function application open parentheses A divided by 2 close parentheses cot invisible function application open parentheses B divided by 2 close parentheses minus 1 over denominator cot invisible function application open parentheses A divided by 2 close parentheses plus cot invisible function application open parentheses B divided by 2 close parentheses end fraction$
$equals fraction numerator fraction numerator 6 over denominator 5 end fraction cross times fraction numerator 37 over denominator 20 end fraction minus 1 over denominator fraction numerator 6 over denominator 5 end fraction plus fraction numerator 37 over denominator 20 end fraction end fraction$
$equals fraction numerator 222 minus 100 over denominator 120 plus 185 end fraction equals fraction numerator 122 over denominator 305 end fraction equals fraction numerator 2 over denominator 5 end fraction$
Also, $tan invisible function application fraction numerator A over denominator 2 end fraction tan invisible function application fraction numerator C over denominator 2 end fraction equals square root of fraction numerator open parentheses s minus b close parentheses open parentheses s minus c close parentheses over denominator s open parentheses s minus a close parentheses end fraction end root blank square root of fraction numerator open parentheses s minus a close parentheses blank open parentheses s minus b close parentheses over denominator s open parentheses s minus c close parentheses end fraction end root$
$rightwards double arrow fraction numerator 5 over denominator 6 end fraction cross times fraction numerator 2 over denominator 5 end fraction equals fraction numerator s minus b over denominator s end fraction$
$rightwards double arrow 3 open parentheses s minus b close parentheses equals s$
$rightwards double arrow 2 s equals 3 b blank rightwards double arrow a plus b plus c equals 3 b$
$rightwards double arrow a plus c equals 2 b$
$rightwards double arrow a comma blank b blank$ and $blank c blank$ are in A.P.

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