Question

In given figure minimum distance to reach from point C to point A will be

A.$$a^2$$
B.$$\sqrt{2}$$
C.$$2$$
D.$$2a$$

Answer 1

Hint: Given is a right angled triangle. So use measurements of sides opposite to $${60}^{\circ }$$ and $${30}^{\circ }$$ remaining angles of triangles using $${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$$ triangle theorem.

Complete step-by-step answer:
Given figure is a right angled triangle.
One angle measures $${60}^{\circ }$$ then remaining should be undoubtedly$${30}^{\circ }$$ because the sum of all angles of a triangle is $${180}^{\circ }$$.
From $${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$$ triangle theorem,
Side opposite to $${30}^{\circ }$$= $${\displaystyle \frac{1}{2}}\times hypotenuse$$
Side opposite to $${60}^{\circ }$$ =$${\displaystyle \frac{\sqrt{3}}{2}}\times hypotenuse$$
Thus the minimum distance to reach point A from point C is AC only that is hypotenuse of the figure. But we have no length given .But we have a side opposite to hypotenuse is given that is equal to a.
Now let’s use our formula.
Side opposite to $${30}^{\circ }$$= $${\displaystyle \frac{1}{2}}\times hypotenuse$$
$$\begin{gathered} \Rightarrow a={\displaystyle \frac{1}{2}}\times hypotenuse \\ \Rightarrow hypotenuse=2a \end{gathered}$$
This is the minimum distance $$2a$$.
Thus, option D is correct.
Note: In a right angled triangle angles are$${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$$.
Two angles are already given and only need to find the remaining.
Sides opposite to respective angles have measures related with hypotenuse.
In a right angled triangle we generally use Pythagoras’s theorem to find hypotenuse unless measures of base and height of the triangle are given.