Question

line segment yv of rectangle yvwx measures 24 units. what is the length of line segment yx?
○ 8 units
○ 8√3 units
○ 12 units
○ 12√3 units

Explanation:

Step1: Identify triangle properties

In rectangle YVWX, triangle XVW is a right triangle with $\angle VXW = 30^\circ$, $\angle XVW = 60^\circ$, and $YV = XW = 24$ (opposite sides of rectangle). Wait, no, YV is 24, and in triangle, we can use trigonometry. Wait, actually, in triangle XVW, $\angle XWV = 90^\circ$, $\angle VXW = 30^\circ$, so the side opposite $30^\circ$ is VW, and hypotenuse XV? Wait, no, YV is 24, which is equal to XW (rectangle opposite sides). Wait, maybe better: in triangle, $\cos(30^\circ)=\frac{XW}{XV}$? No, wait, YV is 24, which is the length of YV, so YV is equal to XW (since YVWX is rectangle, YV || XW and YV = XW). So XW = 24. In triangle XWV, $\angle VXW = 30^\circ$, $\angle XWV = 90^\circ$, so $\tan(30^\circ)=\frac{VW}{XW}$? No, wait, we need YX. YX is equal to VW (opposite sides of rectangle). So in triangle XWV, $\angle VXW = 30^\circ$, XW = 24, and we can use $\tan(30^\circ)=\frac{VW}{XW}$? Wait, no, $\tan(30^\circ)=\frac{opposite}{adjacent}=\frac{VW}{XW}$? Wait, no, $\angle VXW = 30^\circ$, so the side opposite is VW, adjacent is XW. Wait, but maybe better: in a 30-60-90 triangle, the sides are in ratio $1 : \sqrt{3} : 2$. The side opposite 30° is the shortest, then opposite 60° is $\sqrt{3}$ times that, hypotenuse is 2 times. Wait, here, XW is 24, which is adjacent to 30°, so adjacent side (XW) is opposite 60°? Wait, no, let's label the triangle: angle at X is 30°, angle at W is 90°, so angle at V is 60°. So side XW is adjacent to 30°, side VW is opposite to 30°, and side XV is hypotenuse. So in 30-60-90 triangle, adjacent to 30° (XW) is length $x\sqrt{3}$, opposite to 30° (VW) is length $x$, hypotenuse is $2x$. Wait, no, standard 30-60-90: side opposite 30° is $x$, opposite 60° is $x\sqrt{3}$, hypotenuse $2x$. So if angle at X is 30°, then side opposite (VW) is $x$, side adjacent (XW) is $x\sqrt{3}$, hypotenuse (XV) is $2x$. We know XW = 24 (since YV = 24 and YV = XW in rectangle). So $x\sqrt{3}=24$? No, wait, no: XW is adjacent to 30°, so adjacent side is $x\sqrt{3}$ (opposite 60°), and opposite side (VW) is $x$. Wait, no, let's correct: in triangle, angle at X: 30°, angle at W: 90°, so angle at V: 60°. So side opposite 30° (angle X) is VW, side opposite 60° (angle V) is XW, side opposite 90° (angle W) is XV. So by 30-60-90 ratios: VW (opposite 30°) = $x$, XW (opposite 60°) = $x\sqrt{3}$, XV (hypotenuse) = $2x$. We know XW = 24, so $x\sqrt{3}=24$? No, wait, XW is opposite 60°, so XW = $x\sqrt{3}$, and VW (which is equal to YX, since YX and VW are opposite sides of rectangle) is $x$. So we need to find YX = VW = $x$. Given XW = 24 = $x\sqrt{3}$, so $x = \frac{24}{\sqrt{3}} = 8\sqrt{3}$? Wait, no, that can't be. Wait, maybe I mixed up the angles. Wait, the angle at V is 60°, so angle at V: 60°, angle at W: 90°, so angle at X: 30°. So side opposite angle X (30°) is VW, side opposite angle V (60°) is XW, side opposite angle W (90°) is XV. So ratios: VW (30° opposite) = $x$, XW (60° opposite) = $x\sqrt{3}$, XV (hypotenuse) = $2x$. So XW = $x\sqrt{3} = 24$, so $x = \frac{24}{\sqrt{3}} = 8\sqrt{3}$? But that's not one of the options? Wait, no, wait the options are 8, 8√3, 12, 12√3. Wait, maybe I made a mistake. Wait, YV is 24, which is the length of YV, so YV is the top side, so YV = XW = 24? Wait, no, in the rectangle, YV and XW are opposite sides, so YV = XW. YX and VW are opposite sides, so YX = VW. Now, in triangle XVW, angle at X is 30°, angle at W is 90°, so it's a 30-60-90 triangle. The hypotenuse is XV, and the side adjacent to 30° is XW (length 24), and the side opposite to 30° is VW (length YX). Wait, no, adjacent to 30° is XW, opposite is VW. So $\sin(30^\circ) = \frac{VW}{XV}$, $\cos(30^\circ) = \frac{XW}{XV}$. Wait, maybe better to use $\tan(30^\circ) = \frac{VW}{XW}$. So $\tan(30^\circ) = \frac{YX}{24}$. Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, so $YX = 24 \times \frac{1}{\sqrt{3}} = \frac{24}{\sqrt{3}} = 8\sqrt{3}$. Wait, but 8√3 is an option. Wait, but let me check again. Wait, maybe the angle is at V: 60°, so angle at V is 60°, so in triangle, angle at V is 60°, angle at W is 90°, so angle at X is 30°. So side XW is opposite angle V (60°), so XW = XV × sin(60°), and YX (VW) is opposite angle X (30°), so YX = XV × sin(30°). But we know YV = 24, which is equal to XW (since YV and XW are opposite sides of rectangle). So XW = 24 = XV × sin(60°). So XV = 24 / sin(60°) = 24 / (√3/2) = 48 / √3 = 16√3. Then YX = XV × sin(30°) = 16√3 × 1/2 = 8√3. Yes, that matches. So YX is 8√3 units.

Step2: Confirm with trigonometry

In right triangle XVW, $\angle XWV = 90^\circ$, $\angle VXW = 30^\circ$, $XW = 24$ (since $YV = XW$ in rectangle). Using $\tan(30^\circ) = \frac{YX}{XW}$ (as $YX = VW$), $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. So $YX = XW \times \tan(30^\circ) = 24 \times \frac{1}{\sqrt{3}} = 8\sqrt{3}$.

Answer:

$8\sqrt{3}$ units (corresponding to the option "8√3 units")