General
General
Easy
Question
Find the values of x and y.
If your answers are not integers, it should be in simplest radical form.
Hint:
We are given a right-angled triangle. We are given one of its angle. It is 30°. The other angle will be 60°. We are given the values of one of its side. It is 12. We are asked to find the values of the remaining sides.
The correct answer is:
Let the given triangle be ABC.
∠ABC = 90°
∠BCA = 30°
AB = 12
BC = y
AC = x
The sum of all angles of a triangle is 180°. Therefore, the remaining angle will be 60°
So, ∠BAC = 60°
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 BC. And the side opposite to 60° is AB.
Length of smallest side = y
Hypotenuse = x
Length of longer side = 12
So we can write,
Length of longer leg = √3 ( length of smaller leg)
AB = √3(BC)
So, to find the smaller leg we have to divide the longer side by √3.
BC = AB ÷ √3
BC = 12 ÷ √3
y = 4√3
Now,
Hypotenuse = 2(length of smallest side)
AC = 2(4√3)
x = 8√3
Therefore, the length of the hypotenuse is 8√3.
The values x = 8√3 and y = 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 and trigonometric ratios.
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