Question

In a right triangle $AC=BC$ and $D$ is the midpoint of $AC$, then cotangent of angle $DBC$ is equal to
A. $2$
B. $3$
C. $\frac{1}{2}$
D. $\frac{1}{3}$

Answer 1

Hint: To solve this question, we will consider the triangle $DBC$. In triangle $DBC$, we will calculate the cotangent of triangle $DBC$ that is $\cot B$ using trigonometric ratio formula and midpoint properties.
A midpoint divides the length into equal parts.
Formula used:
The trigonometric formula of cotangent is,
$\cot \theta =\frac{Base}{Perpendicular}$
Complete step-by-step solution:
We are given a right angled triangle in which two sides are equal that is $AC=BC$ and $D$ is the midpoint of side $AC$. We have to calculate the cotangent of angle $DBC$.
We will draw diagram for a triangle with the given data.

As $D$ is the midpoint of side$AC$, then$AD=DC$.
Now in triangle $DBC$,
$\cot B=\frac{BC}{DC}$
As $AC=BC$ and$AD=DC$, we can write $BC$ as $BC=2DC$.
$\begin{align}
  & \cot B=\frac{2DC}{DC} \\
 & \cot B=2 \\
\end{align}$
The cotangent of angle $DBC$is equal to $\cot B=2$ in a right angled triangle when$AC=BC$ and $D$ is the midpoint of $AC$.Hence the correct option is (A).
Note:
We could have also calculated the tangent of angle $DBC$ and then inversed the result derived because we know that cotangent is the inverse of tangent that is $\cot \theta =\frac{1}{\tan \theta }$.
A right angled triangle is a type of triangle in which one angle is always ${{90}^{0}}$. The three sides of a right angled triangle are termed as base, perpendicular or altitude and hypotenuse where hypotenuse is the longest side and it is the side opposite to the right angle that is ${{90}^{0}}$. The relationship between the sides of this triangle is depicted by the Pythagoras theorem which is ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$.