In a triangle $A B C,$ the median $A D$ is perpendicular to $A C .$ If $b = 5 c m$ and $c = 8 c m,$ then $a$ is equal to ?

$10$

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$12$

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$14$

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$\sqrt{221}$

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Solution

The correct option is C
$14$

Step 1: Illustrating the given information
$A B C$ is a triangle in which $A D$ is perpendicular to $A C .$
$A B = c, B C = a, A C = b$
$b = 5 c m$ and $c = 8 c m$
Step 2: Applying trigonometric ratios in the triangle:
$\cos (C) = \frac{A C}{C D} \Rightarrow \cos (C) = \frac{5}{\frac{B C}{2}} (G i v e n : B C = 2 C D = 2 B D) \Rightarrow \cos (C) = \frac{10}{B C} = \frac{10}{a}$
Also, $\cos (C) = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$
Step 3: Equating the two equations:
$\frac{a^{2} + b^{2} - c^{2}}{2 a b} = \frac{10}{a} \Rightarrow a^{2} + b^{2} - c^{2} = 20 b \Rightarrow a^{2} + 25 - 121 = 100 \Rightarrow a^{2} = 196 \Rightarrow a = 14 c m$

So, the value of $a$ is $14 c m .$
Therefore, Option (C) is correct.

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