Question

please answer this question and also give all steps. question is in image
Options are -:
A) 30°
B) 60°
C) 45°
D) 90°​

Answer 1

Answer: B) 60°

Given that l₃ is the diameter of the circumcircle of the triangle of sides l₁, l₂ and l₃.

  ⇒  l₃ is one side of the triangle.

  ⇒  l₃ is the diameter of the circumcircle of the triangle.

If one side of a triangle is the diameter of its circumcircle, then the triangle will be a right triangle, and the diameter will be the hypotenuse of the right triangle.

Thus, the triangle is right triangle and l₃ is the hypotenuse. So angle between 1₁ and l₂ is 90°.

Given that l₃ = 2l₁.

Aren't we familiar to any triangle where one side is twice the other? If there is, what about the angles of the triangle?

What about triangles with angles 30°, 60° and 90°?!

Okay, recall the ratio of corresponding sides of the 30°, 60°, 90° triangle.

$\begin{tabular}{c|c|c}30\textdegree&60\textdegree&90\textdegree\\ \cline{1-3}1&\sqrt{3}&2\\ \cline{1-3} l_1 & l_2 & 2 l_1 \\ \cline{1-3} l_1 & l_2 & l_3 \end{tabular}$

The question is to find the angle between l₁ and l₃, i.e., angle opposite to l₂.

In the table it seems that the angle opposite to l₂ is 60°, thus 60° is the answer.

Or see the following:

l₃ = 2l₁

l₁ / l₃  =  1/2

Here seems that the ratio of l₁ and l₃ is 1/2. Which trigonometric ratio is this? Isn't it sin 30°? Or cos 60°?

Taking cos 60° for better.

As we said earlier, we have to find the angle opposite to l₂. When we take cos 60° = l₁ / l₃ = 1/2,  as l₁ is the adjacent side of this 60°, thus l₂ is the opposite side. Hence we get here too that 60° is the answer.

Thus option (B) is the answer.