Question

g.gsr.6.1 (lc)
find the values of x and y.
6
60°
x
у
30°
o x = 3√3, y = 12
o x = 6√3, y = 12
o x = 12, y = 6√3
o x = 6, y = 6√2

Explanation:

Step1: Identify Triangle Type

This is a 30-60-90 right triangle. In such a triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\) (short leg : long leg : hypotenuse). The side opposite \(30^\circ\) is the short leg, opposite \(60^\circ\) is the long leg, and hypotenuse is opposite \(90^\circ\). Here, the side with length 6 is opposite \(30^\circ\)? Wait, no—wait, the angle of \(30^\circ\) is at the bottom, \(60^\circ\) at the top left, right angle at top right. So the side adjacent to \(60^\circ\) (the horizontal side) is 6. Wait, let's reorient: the right angle is between the horizontal side (length 6) and the vertical side \(x\). The angle at the top left is \(60^\circ\), so the angle at the bottom is \(30^\circ\). So the side opposite \(30^\circ\) is the horizontal side (length 6), so short leg is 6. Then hypotenuse \(y\) is twice the short leg: \(y = 2 \times 6 = 12\). The long leg \(x\) (opposite \(60^\circ\)) is short leg times \(\sqrt{3}\): \(x = 6\sqrt{3}\).

Step2: Verify with Ratios

In 30-60-90 triangle:

• Short leg (opposite \(30^\circ\)): \(a\)
• Long leg (opposite \(60^\circ\)): \(a\sqrt{3}\)
• Hypotenuse: \(2a\)

Here, short leg is 6 (opposite \(30^\circ\)), so:

• Hypotenuse \(y = 2 \times 6 = 12\)
• Long leg \(x = 6 \times \sqrt{3} = 6\sqrt{3}\)

So \(x = 6\sqrt{3}\), \(y = 12\), which matches the second option.

Answer:

\(x = 6\sqrt{3}\), \(y = 12\) (corresponding to the option "x = 6√3, y = 12")