Question

What is the missing side length x and y?
Imagine a right triangle. The left side is labelled x, the bottom side is labelled y, and the slanted side on the right, is labelled 10.
In the bottom right corner of the triangle is 60 degrees, and in the top corner is 30 degrees.

Answer 1

In a right triangle, you can use the properties of the angles and the sides to find the lengths of the sides. In this case, you have a right triangle with one angle measuring \(60^\circ\), another measuring \(30^\circ\), and the hypotenuse (the slanted side) measuring 10.

In a 30-60-90 triangle, the sides are in a specific ratio:

- The side opposite the \(30^\circ\) angle is half the length of the hypotenuse.
- The side opposite the \(60^\circ\) angle is \(\sqrt{3}\) times the length of the side opposite the \(30^\circ\) angle.

1. **Finding side opposite to \(30^\circ\) (y):**
\[
y = \frac{1}{2} \times \text{hypotenuse} = \frac{1}{2} \times 10 = 5
\]

2. **Finding side opposite to \(60^\circ\) (x):**
\[
x = y \times \sqrt{3} = 5 \times \sqrt{3} \approx 5 \times 1.732 = 8.66
\]

So, the side lengths are:
- \(y = 5\)
- \(x \approx 8.66\) (or \(5\sqrt{3}\)).

Thus:
- \(x \approx 8.66\)
- \(y = 5\)