Question

14 find the values of x and y. triangle image with right angle, side 12, angle 60°, angle 30°, sides x and y eave in simplest radical form, if necessary.

Explanation:

Step1: Identify Triangle Type

The triangle is a right - angled triangle with angles \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).

Looking at the triangle, the side of length \(12\) is opposite the \(60^{\circ}\) angle? Wait, no. Wait, the right angle, one angle is \(60^{\circ}\), one is \(30^{\circ}\). Let's check the sides. The side with length \(12\): let's see, if we consider the side opposite \(30^{\circ}\) as \(x\)? Wait, no. Wait, the side adjacent to \(60^{\circ}\) and opposite \(30^{\circ}\)? Wait, let's re - analyze.

In a \(30 - 60 - 90\) triangle, the hypotenuse is the longest side. Let's assume that the side of length \(12\) is opposite the \(60^{\circ}\) angle? No, wait, the side opposite \(30^{\circ}\) is the shortest. Wait, maybe the side of length \(12\) is the side opposite \(60^{\circ}\), and \(x\) is opposite \(30^{\circ}\), and \(y\) is the hypotenuse.

The ratio of sides in a \(30 - 60 - 90\) triangle: if the side opposite \(30^{\circ}\) is \(a\), side opposite \(60^{\circ}\) is \(a\sqrt{3}\), hypotenuse is \(2a\).

Let's suppose that the side of length \(12\) is the side opposite \(60^{\circ}\), so \(a\sqrt{3}=12\), then \(a=\frac{12}{\sqrt{3}} = 4\sqrt{3}\). Wait, no, maybe I got it wrong. Wait, the side with length \(12\) is adjacent to \(60^{\circ}\) and opposite \(30^{\circ}\)? No, the right angle is between \(x\) and \(12\), so \(x\) and \(12\) are the legs, and \(y\) is the hypotenuse.

Angle of \(30^{\circ}\): the side opposite \(30^{\circ}\) is \(x\), the side opposite \(60^{\circ}\) is \(12\), and hypotenuse is \(y\).

In a \(30 - 60 - 90\) triangle, \(\tan(60^{\circ})=\frac{12}{x}\), since \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). \(\tan(60^{\circ})=\sqrt{3}\), so \(\sqrt{3}=\frac{12}{x}\), then \(x = \frac{12}{\sqrt{3}}=\frac{12\sqrt{3}}{3}=4\sqrt{3}\)? Wait, no, maybe I mixed up the angles. Wait, the angle of \(30^{\circ}\): \(\tan(30^{\circ})=\frac{x}{12}\), since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), so \(\frac{1}{\sqrt{3}}=\frac{x}{12}\), then \(x = \frac{12}{\sqrt{3}} = 4\sqrt{3}\). And the hypotenuse \(y\): using Pythagoras, \(y=\sqrt{x^{2}+12^{2}}=\sqrt{(4\sqrt{3})^{2}+12^{2}}=\sqrt{48 + 144}=\sqrt{192}=8\sqrt{3}\)? Wait, no, that can't be. Wait, maybe the side of length \(12\) is the side opposite \(30^{\circ}\). Wait, if the side opposite \(30^{\circ}\) is \(12\), then the hypotenuse \(y = 2\times12=24\), and the side opposite \(60^{\circ}\) (which is \(x\)) is \(12\sqrt{3}\). Wait, that makes more sense. Let's check the angles. The side opposite \(30^{\circ}\) is the shortest side. So if the side of length \(12\) is opposite \(30^{\circ}\), then:

- Hypotenuse (\(y\)): In a \(30 - 60 - 90\) triangle, hypotenuse \(= 2\times\) (side opposite \(30^{\circ}\)). So \(y = 2\times12=24\).

- Side opposite \(60^{\circ}\) (\(x\)): In a \(30 - 60 - 90\) triangle, side opposite \(60^{\circ}=\sqrt{3}\times\) (side opposite \(30^{\circ}\)). So \(x = 12\sqrt{3}\). Wait, but wait, in the diagram, the side of length \(12\) is adjacent to the \(60^{\circ}\) angle? Wait, maybe I made a mistake in identifying which angle is which. Let's look at the triangle again. The right angle is at the top - left, one angle at the top - right is \(60^{\circ}\), one at the bottom is \(30^{\circ}\). So the side of length \(12\) is adjacent to the \(30^{\circ}\) angle and opposite the \(60^{\circ}\) angle? Wait, no. Let's use trigonometric ratios.

Using trigonometry:

\(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{x}{12}\)

Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), we have \(\frac{1}{\sqrt{3}}=\frac{x}{12}\), so \(x=\frac{12}{\sqrt{3}} = 4\sqrt{3}\) (rationalizing the denominator: \(\frac{12\sqrt{3}}{3}=4\sqrt{3}\))

\(\cos(30^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{12}{y}\)

Since \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{12}{y}\), so \(y=\frac{24}{\sqrt{3}} = 8\sqrt{3}\)? Wait, no, that contradicts the \(30 - 60 - 90\) ratio. Wait, I think I messed up the trigonometric ratios. Let's correct:

In a right - angled triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\)

For the \(30^{\circ}\) angle at the bottom:

- Opposite side: \(x\)

- Adjacent side: \(12\)

So \(\tan(30^{\circ})=\frac{x}{12}\), \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), so \(x = \frac{12}{\sqrt{3}}=4\sqrt{3}\)

For the hypotenuse \(y\), using Pythagoras: \(x^{2}+12^{2}=y^{2}\)

Substitute \(x = 4\sqrt{3}\): \((4\sqrt{3})^{2}+12^{2}=y^{2}\)

\(16\times3 + 144=y^{2}\)

\(48+144=y^{2}\)

\(192=y^{2}\), so \(y=\sqrt{192}=\sqrt{64\times3}=8\sqrt{3}\)

Wait, but this is conflicting with the \(30 - 60 - 90\) ratio. Wait, maybe the side of length \(12\) is the hypotenuse? No, the hypotenuse is the longest side. Wait, no, in the diagram, the side of length \(12\) is one of the legs. Wait, maybe the angle at the top - right is \(30^{\circ}\) and at the bottom is \(60^{\circ}\). Let's try that.

If the angle at the top - right is \(30^{\circ}\) and at the bottom is \(60^{\circ}\):

- \(\tan(60^{\circ})=\frac{12}{x}\), \(\tan(60^{\circ})=\sqrt{3}\), so \(\sqrt{3}=\frac{12}{x}\), \(x=\frac{12}{\sqrt{3}} = 4\sqrt{3}\)

- \(\cos(60^{\circ})=\frac{x}{y}\), \(\cos(60^{\circ})=\frac{1}{2}\), so \(\frac{1}{2}=\frac{4\sqrt{3}}{y}\), \(y = 8\sqrt{3}\). No, this is the same as before.

Wait, I think the initial mistake was in the ratio. Let's recall: in a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is \(s\), side opposite \(60^{\circ}\) is \(s\sqrt{3}\), hypotenuse is \(2s\).

Looking at the triangle, if the side opposite \(30^{\circ}\) is \(x\), side opposite \(60^{\circ}\) is \(12\), and hypotenuse is \(y\). Then:

\(s\sqrt{3}=12\) (since side opposite \(60^{\circ}\) is \(s\sqrt{3}\)), so \(s=\frac{12}{\sqrt{3}} = 4\sqrt{3}\) (this is \(x\), the side opposite \(30^{\circ}\))

Hypotenuse \(y = 2s=2\times4\sqrt{3}=8\sqrt{3}\)? No, that can't be, because \(8\sqrt{3}\approx13.85\), which is longer than \(12\) but shorter than if \(y = 24\). Wait, I think the problem is in identifying which side is which.

Wait, let's start over. Let's assume that the side of length \(12\) is the side opposite \(30^{\circ}\). Then:

- Side opposite \(30^{\circ}\): \(12\)

- Side opposite \(60^{\circ}\) (\(x\)): \(12\sqrt{3}\)

- Hypotenuse (\(y\)): \(2\times12 = 24\)

Now, let's check with trigonometry:

\(\sin(30^{\circ})=\frac{12}{y}\), \(\sin(30^{\circ})=\frac{1}{2}\), so \(\frac{1}{2}=\frac{12}{y}\), \(y = 24\) (correct, since \(\sin(30^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\))

\(\sin(60^{\circ})=\frac{x}{y}\), \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\), \(y = 24\), so \(x=24\times\frac{\sqrt{3}}{2}=12\sqrt{3}\)

Ah! So I had the angle identification wrong earlier. The side of length \(12\) is opposite the \(30^{\circ}\) angle, not the \(60^{\circ}\) angle. That's the key mistake. So:

- Side opposite \(30^{\circ}\): \(12\) (so this is the shortest side)

- Hypotenuse (\(y\)): \(2\times12 = 24\) (since hypotenuse \(= 2\times\) side opposite \(30^{\circ}\) in \(30 - 60 - 90\) triangle)

- Side opposite \(60^{\circ}\) (\(x\)): \(\sqrt{3}\times12=12\sqrt{3}\) (since side opposite \(60^{\circ}=\sqrt{3}\times\) side opposite \(30^{\circ}\) in \(30 - 60 - 90\) triangle)

Step2: Calculate \(x\) and \(y\)

- Calculating \(y\) (hypotenuse):
In a \(30 - 60 - 90\) triangle, hypotenuse \(= 2\times\) (side opposite \(30^{\circ}\)). The side opposite \(30^{\circ}\) is \(12\), so \(y = 2\times12=24\).

- Calculating \(x\) (side opposite \(60^{\circ}\)):
In a \(30 - 60 - 90\) triangle, side opposite \(60^{\circ}=\sqrt{3}\times\) (side opposite \(30^{\circ}\)). The side opposite \(30^{\circ}\) is \(12\), so \(x = 12\sqrt{3}\).

Answer:

\(x = 12\sqrt{3}\), \(y = 24\)