Question

what is the value of x? 30° x 18

Explanation:

Step1: Identify triangle type

It's a right - triangle with a 30° angle. In a 30 - 60 - 90 triangle, the side opposite 30° is half the hypotenuse, and the side opposite 60° is $\frac{\sqrt{3}}{2}$ times the hypotenuse. Here, the side of length 18 is adjacent to the 30° angle, and \(x\) is opposite the 30° angle? Wait, no. Wait, let's re - examine. The right angle, a 30° angle, so the sides: let's label the triangle. Let the right angle be \(C\), the 30° angle be \(A\), so side \(a\) (opposite \(A\)) is \(x\), side \(b = 18\) (adjacent to \(A\)), hypotenuse \(c\). Wait, in a right - triangle, \(\sin(30^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos(30^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan(30^{\circ})=\frac{\text{opposite}}{\text{adjacent}}\). Wait, actually, if the angle is 30°, and the side adjacent to 30° is 18? No, wait, maybe the side of length 18 is the hypotenuse? Wait, no, in a 30 - 60 - 90 triangle, the side opposite 30° is the shortest side, half the hypotenuse. Wait, let's look at the triangle again. The right angle, a 30° angle, so the sides: the side with length 18 – is it the hypotenuse or one of the legs? Wait, the angle of 30° is at the top, the right angle is at the bottom right. So the side labeled 18 is adjacent to the 30° angle? No, wait, the side labeled 18 is one of the legs, and \(x\) is the other leg? Wait, no, let's use trigonometry. \(\sin(30^{\circ})=\frac{x}{18}\)? Wait, no, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). Wait, maybe the side of length 18 is the hypotenuse? Wait, no, if the angle is 30°, and the side opposite 30° is \(x\), and the hypotenuse is 18? No, that can't be, because in a 30 - 60 - 90 triangle, the side opposite 30° is half the hypotenuse. Wait, I think I made a mistake. Let's start over. In a right - triangle, if one angle is 30°, then the side opposite the 30° angle is half the hypotenuse. Wait, no, the side opposite 30° is the shortest side, and it's half the hypotenuse. So if the side opposite 30° is \(x\), and the hypotenuse is \(h\), then \(x=\frac{h}{2}\). But here, the side of length 18: is it the hypotenuse or the side adjacent to 30°? Wait, looking at the triangle, the side with length 18 is adjacent to the 30° angle? No, the side with length 18 is one of the legs, and \(x\) is the other leg. Wait, no, let's use trigonometric ratios. \(\tan(30^{\circ})=\frac{x}{18}\)? No, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So if the angle is 30°, the opposite side is \(x\), adjacent side is 18, then \(\tan(30^{\circ})=\frac{x}{18}\), so \(x = 18\times\tan(30^{\circ})\). But \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx0.577\), but that doesn't seem right. Wait, maybe the side of length 18 is the hypotenuse. Then the side opposite 30° is \(x=\frac{18}{2}=9\)? Wait, that makes sense. Because in a 30 - 60 - 90 triangle, the side opposite 30° is half the hypotenuse. So if the hypotenuse is 18, then \(x = 9\). Wait, but why is the side of length 18 the hypotenuse? Let's check the triangle structure. The right angle is at the bottom right, the 30° angle is at the top, so the side connecting the top to the right angle is \(x\) (opposite 30°), the side connecting the top to the bottom left is 18 (hypotenuse), and the side connecting bottom left to bottom right is the other leg. Yes, that makes sense. So hypotenuse is 18, angle at top is 30°, so side opposite 30° (which is \(x\)) is \(\frac{1}{2}\times\) hypotenuse. So \(x=\frac{18}{2}=9\). Wait, but let's confirm with trigonometry. \(\sin(30^{\circ})=\frac{x}{18}\), and \(\sin(30^{\circ}) = 0.5\), so \(x=18\times0.5 = 9\). Yes, that's correct.

Step2: Calculate \(x\)

Using the property of 30 - 60 - 90 triangles: the side opposite the 30° angle is half the hypotenuse. Here, the hypotenuse is 18, so the side opposite 30° (which is \(x\)) is \(\frac{18}{2}=9\).

Answer:

\(9\)