Question

From the given figure, find the angle of depression of point C from point P .
     

(A). ${{90}^{o}}$
(B). ${{60}^{o}}$
(C). ${{30}^{o}}$
(D). ${{45}^{o}}$

Answer 1

Hint: The given problem is related to the angle of depression. Use the definition of angle of depression to identify the angle to be calculated. Then, use the angle sum property of the triangle to calculate the value of the required angle.

Complete step-by-step answer:

The term angle of depression denotes the angle from the horizontal downward to an object. So, the angle of depression of point C from point P will be $\angle $QPC. Let it be \[\theta \].
Now, we know that the angle sum property of a triangle states that the sum of interior angles of a triangle is equal to ${{180}^{o}}$.
So, in ∆ APB, $\angle $ A +$\angle $ APB +$\angle $ ABP = ${{180}^{o}}$
$\Rightarrow {{90}^{o}}$ + $\angle $ APB + ${{60}^{o}}$ = ${{180}^{o}}$
$\Rightarrow {{150}^{o}}+\angle APB={{180}^{o}}$
$\Rightarrow \angle $ APB = ${{30}^{o}}$
Also, $\angle $ P = ${{90}^{o}}$. (right angle)
So, $\angle $APB +$\angle $BPC + $\angle $QPC = ${{90}^{o}}$
$\Rightarrow {{30}^{o}}$ + ${{30}^{o}}$ + \[\theta \] = ${{90}^{o}}$
\[\Rightarrow \theta \] = ${{30}^{o}}$
Hence, the value of the angle of depression of point C from point P comes out to be ${{30}^{o}}$ .

Note: In this question, one must take care to take the depression angle from the horizontal. Students generally make mistakes in recognizing the angle of depression.

If from point ‘O’, we are watching point ‘A’, angle ‘ $\phi $ ’ is the angle of depression. But students get confused and take the angle ‘ $\alpha $ ’ as the angle of depression. Such mistakes should be avoided as due to these mistakes, students can end up getting a wrong answer. The definition of angle of depression and angle of elevation should be properly understood to avoid such mistakes.