In the following figure, AD bisects angle BAC. Find the length (in cm) of BD.

4

According to the concepts

\(\frac{6}{2x\;-\;3}\) = \(\frac{x\;-\;2}{x}\)

⇒ 6x = 2\( { x}^{2 } \) - 4x - 3x + 6

⇒ 6x = 2\( { x}^{2 } \) - 7x + 6

⇒ 0 = 2\( { x}^{2 } \) - 7x + 6 - 6x

⇒ 2\( { x}^{2 } \) - 7x + 6 = 0

⇒ 2\( { x}^{2 } \) - 12x - x + 6 = 0

⇒ 2x(x - 6) - (x - 6) = 0

⇒ (x - 6)(2x -1) = 0

So, x = 6, \(\frac{1}{2}\)

For x = \(\frac{1}{2}\)

BD = \(\frac{1}{2}\) - 2

⇒ -\(\frac{3}{2}\) as the value of a side cannot be negative

So, x = 6

Now,

BD = 6 - 2

⇒ 4 cm

The length of BD is 4 cm.