Question
Find the values of a and b.
If your answers are not integers, it should be in simplest radical form.
- a = 22 b = 11
- a = 11 b = 22
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 11√3. We are asked to find the values of the remaining sides.
The correct answer is: a = 22 b = 11
Let the given triangle be ABC.
∠ABC = 90°
∠BAC = 30°
AB = 11√3
BC = b
AC = a
The sum of all angles of a triangle is 180°. Therefore, the remaining angle will be 60°
So, ∠BCA = 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 = b
Hypotenuse = a
Length of longer side = 11√3
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 = 11√3 ÷ √3
b = 11
Now,
Hypotenuse = 2(length of smallest side)
AC = 2(11)
a = 22
Therefore, the length of the hypotenuse is 22
The values a = 22 and b = 11
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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