Question

In the below figure, AD = 4cm, BD = 3cm and CB = 12cm, find cot θ:

a)$\dfrac{12}{5}$
b)$\dfrac{5}{12}$
c)$\dfrac{13}{12}$
d)$\dfrac{12}{13}$

Answer 1

Hint: For the value of cot θ, we need the length of the side AB because$\cot \theta =\dfrac{CB}{AB}$. In$\Delta ADB$, AB is the hypotenuse of the triangle and we know the lengths of AD and BD so using Pythagoras theorem we can find the length of the side AB. Now, substitute this value of AB in the cot θ equation.

Complete step-by-step answer:
The figure given below shows two right angled triangles$\Delta ABC\And \Delta ADB$in which AD = 4cm, BD = 3cm and CB = 12cm and$\angle ACB=\theta $.

We are asked to find the value of cot θ and we know that,
$\cot \theta =\dfrac{CB}{AB}$ ………… Eq. (1)

The value of CB is given in the question and AB is unknown so we have to find the value of AB.

If you look carefully at$\Delta ADB$you will find that AB is the hypotenuse of this triangle so using Pythagoras theorem we can find the length of side AB.

$\begin{align}

  & A{{B}^{2}}=A{{D}^{2}}+B{{D}^{2}} \\

 & \Rightarrow A{{B}^{2}}={{4}^{2}}+{{3}^{2}} \\

 & \Rightarrow A{{B}^{2}}=16+9=25 \\

 & \Rightarrow AB=5 \\

\end{align}$

From the above calculation, the length of AB is equal to 5cm.

Now, substituting this value of AB in eq. (1) we get,

$\begin{align}

  & \cot \theta =\dfrac{CB}{AB} \\

 & \Rightarrow \cot \theta =\dfrac{12}{5} \\

\end{align}$

From the above, we have found the value of$\cot \theta =\dfrac{12}{5}$.

Hence, the correct option is (a).


Note: One interesting thing to note that in the above solution is while taking square root of$A{{B}^{2}}=25$we have only taken the positive value i.e. AB = 5 we have ignored the negative value that comes after taking the square root i.e. -5 because the length of the side of a triangle cannot be negative.
The interesting point is that if you would have ignored that negative value of AB still you will get the correct answer but you should know the logic behind ignoring the negative value of AB.