If $CD = 15$, $DB = 9$, $AD$ bisects $\angle A$ , $\angle ABC = 90^\circ $, then $AB$ has length

A) 32
B) 18
C) 7
D) 24

Hint:
Here, we will use the trigonometric identity of a tangent for the two right angle triangles formed by the line bisecting the angle. Then we will solve the equation formed by using the trigonometric identities of the right angle triangle to find the length of the base of the right angle triangle.

Formula Used:
Trigonometric Identity: $\tan 2\theta = \dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}$

Complete step by step solution:
We are given that$CD = 15$,$DB = 9$, $AD$ bisects $\angle A$ , $\angle ABC = 90^\circ $
Let $\angle DAB = \theta $
We are given that $AD$ bisects$\angle A$ , so we get$\angle DAB = \angle DAC = \theta $.
Now,
 $\angle CAB = \angle CAD + \angle DAB$
By substituting$\angle DAB = \angle DAC = \theta $, we get
$ \Rightarrow \angle CAB = \theta + \theta = 2\theta $
Let $AB = x$ is the base of the right angle triangle.
We have $\Delta CBA,\Delta DBA$ be the two right angle triangles.
In $\Delta DBA$ , $\tan \theta = \dfrac{{DB}}{{AB}}$
$ \Rightarrow \tan \theta = \dfrac{9}{x}$ ……………………………………………………………………………………………………………$\left( 1 \right)$

In $\Delta CBA$ , $\tan \theta = \dfrac{{CB}}{{AB}}$
We know that $CB = CD + DB = 15 + 9 = 24$
$ \Rightarrow \tan 2\theta = \dfrac{{24}}{x}$
We know that the Trigonometric Identity $\tan 2\theta = \dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}$
By using the trigonometric Identity, we get
$ \Rightarrow \dfrac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }} = \dfrac{{24}}{x}$
By substituting equation $\left( 1 \right)$ , we get
$ \Rightarrow \dfrac{{2\left( {\dfrac{9}{x}} \right)}}{{1 - {{\left( {\dfrac{9}{x}} \right)}^2}}} = \dfrac{{24}}{x}$
By simplifying the equation, we get
$ \Rightarrow \dfrac{{\dfrac{{18}}{x}}}{{1 - \left( {\dfrac{{81}}{{{x^2}}}} \right)}} = \dfrac{{24}}{x}$
$ \Rightarrow \dfrac{{\dfrac{{18}}{x}}}{{\dfrac{{{x^2} - 81}}{{{x^2}}}}} = \dfrac{{24}}{x}$
By cancelling the terms, we get
$ \Rightarrow \dfrac{{18}}{{\dfrac{{{x^2} - 81}}{x}}} = \dfrac{{24}}{x}$
By rewriting the equation, we get
$ \Rightarrow \dfrac{{18x}}{{{x^2} - 81}} = \dfrac{{24}}{x}$
By cross multiplying, we get
$ \Rightarrow 18{x^2} = 24\left( {{x^2} - 81} \right)$
By multiplying the terms, we get
$ \Rightarrow 18{x^2} = 24{x^2} - 1944$
By rewriting the equation, we get
$ \Rightarrow 24{x^2} - 18{x^2} = 1944$
By subtracting the numbers, we get
$ \Rightarrow 6{x^2} = 1944$
Dividing by 6 on both the sides, we get
$ \Rightarrow {x^2} = \dfrac{{1944}}{6}$
$ \Rightarrow {x^2} = 324$
By taking Square root on both the sides, we get
$ \Rightarrow x = \sqrt {324} $
$ \Rightarrow x = 18cm$
$ \Rightarrow AB = 18cm$
Therefore, the length of $AB$ is 18 cm.

Thus, Option (B) is the correct answer.

Note:
We know that Pythagoras theorem is used to find any side of the right angle triangle. But this is applicable only when any two sides of the right angle triangle are known. We are given with only one side of the triangle and one angle, we are using the trigonometric identities to find the unknown side with the given angles. We should know that when a line bisects an angle then the angle is divided into two equal halves.