Find a in this 30-60-90 triangle.

The correct answer is: a = 14

Let the given triangle be ABC.
∠ABC = 90°
∠BAC = 30°
AB = b
BC = 7√3
AC = a
The sum of all angles of a triangle is 180°. Therefore, the remaining angle will be 30°
So, ∠BCA = 30°
It is 30°-60°-90° triangle.
In a 30°-60°-90° triangle, the length of hypotenuse is two times the length of the smallest side. It’s longer side is √3 times the value of the smallest side.
The side opposite to the 30° angle is the smallest side.
The side opposite to the 60° angle is the longer side.
In the given question, the side opposite to 30° is AB.
Length of smallest side = b
Hypotenuse = a
The length of longer side is 7√3
So we can write,
Length of longer leg = √3 ( length of smaller leg)
BC = √3(AB)
So, to find the smaller leg we have to divide the longer side by √3.
AB = BC ÷ √3
AB = 7√3 ÷ √3
b = 7
Now,
Hypotenuse = 2(length of smallest side)
AC = 2(7)
a = 14
Therefore, the length of the hypotenuse is 14.

For such questions, we should know about the properties of a right-angled triangle and 30°-60°-90° triangle. The alternate way to solve the above question is by Pythagoras theorem and trigonometric ratios.