Find the length of u and v in this 30-60-90 triangle.

The correct answers are: u = 8 v = 4√3, u = 4√2 v = 8

Let the given triangle be ABC.
∠ABC = 90°
∠BAC = 60°
AB = 4
BC = v
AC = u
The sum of all angles of a triangle is 180°. Therefore, the remaining angle will be 30°
∠BCA
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 = 4
Hypotenuse = u
The length of longer side is v.
So we can write,
Length of hypotenuse = 2 (length of smallest side)
u = 2(4)
u = 8
Therefore, the length of the hypotenuse is 8.
Length of longer side = √3(length of smallest side)
v = 4√3
Therefore, the length of the longer side is 4√3.
The values of u = 8 and v = 4√3.

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.