Question

In the given figure find the value of x

Answer 1

Let BC = 2y cm

By Apollonius's Theorem

${5}^{2} + {10}^{2} = 2 \times ( {6.5}^{2} + {y}^{2} )$

$\implies \: 25 + 100 = 2( 42.25 + {y}^{2} )$

$\implies \: 2 {y}^{2} = 125 - 84.5$

$\implies \: {y}^{2} = 20.25$

$\implies \: y \: = 4.5$

So the sides are 5 cm, 10cm, 9 cm

$s \: = \frac{(5 + 10 + 9)}{2} = 12$

So

Area

=

$\sqrt{s(s - a \: )(s - b)(s - c)}$

$= \sqrt{12 \times (12 - 4) \times (12 - 10) \times (12 - 9)}$

$= \sqrt{12 \times 7 \times 2 \times 3}$

$= 6\sqrt{14}$

Hence

$6 \sqrt{x} = 6\sqrt{14}$

$\implies \: \sqrt{x} =\sqrt{14}$

$\therefore \: \: x \: = 14$