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Angle in a Semi Circle Is Right Angle

If R is the...

Question

If $R$ is the radius of the circumcircle of the $△ A B C$ and $a (d_{2} + d_{3}) + b (d_{3} + d_{1}) + c (d_{1} + d_{2}) = k R (a + b + c)$ then the value of $^{'} k^{'}$ is

A

1

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B

\frac{1}{2}

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C

\frac{1}{3}

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D

2

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Solution

The correct option is A $1$
Let us consider $△$ ABC to be an equilateral triangle.
So, we get $a = b = c$ and $d = d_{1} = d_{2} = d_{3}$
Also, $a = 2 \sqrt{3} d$ and $R^{2} = d^{2} + (a / 2)^{2}$
Now the whole expression becomes: $2 a d + 2 a d + 2 a d = k R (3 a)$
$\Rightarrow 2 d = k R$
Squaring $\Rightarrow 4 d^{2} = k (d^{2} + \frac{a^{2}}{4})$
$\Rightarrow 4 d^{2} = k (d^{2} + \frac{12 d^{2}}{4})$
$\Rightarrow 4 d^{2} = 4 k d^{2}$
$\Rightarrow k = 1$
So, the correct answer is $1$

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