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Maths-

General

Easy

Question

A circle is inscribed in a triangle ABC touching the side AB at D such that AD = 5, BD = 3. If $straight angle straight A$ = 60° then length BC equals

9
$\frac{120}{13}$
13
12

The correct answer is: 13

$because$ tan 30° = $fraction numerator r over denominator 5 end fraction$
$rightwards double arrow$ r $fraction numerator 5 over denominator square root of 3 end fraction$
Now tan $fraction numerator B over denominator 2 end fraction equals fraction numerator r over denominator 3 end fraction equals fraction numerator 5 over denominator 3 square root of 3 end fraction$ ……..(i)
Now cos B $equals fraction numerator 1 minus tan to the power of 2 end exponent invisible function application fraction numerator B over denominator 2 end fraction over denominator 1 plus tan to the power of 2 end exponent invisible function application fraction numerator B over denominator 2 end fraction end fraction equals fraction numerator 1 over denominator 26 end fraction$
$therefore$ sin B $equals fraction numerator square root of 675 over denominator 26 end fraction equals fraction numerator 15 square root of 3 over denominator 26 end fraction$
$therefore$ sin C = sin(A + B) = sinA cosB + cosAsinB
= $fraction numerator 4 square root of 3 over denominator 13 end fraction$
Now $fraction numerator a over denominator sin invisible function application A end fraction equals fraction numerator c over denominator sin invisible function application C end fraction rightwards double arrow$ a = 13

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