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

Explanation:

Step1: Identify right - triangle properties

In right - triangle YVX, we know that $\angle VXY = 30^{\circ}$ and $\angle VYX=60^{\circ}$, and $YV$ is the side opposite the $30^{\circ}$ angle. Let $YX$ be the hypotenuse.

Step2: Apply 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the side opposite the $30^{\circ}$ angle is $a$, and the hypotenuse is $c$, the ratio of the sides is $a:c = 1:2$ when considering the relationship between the side opposite the $30^{\circ}$ angle and the hypotenuse. Here, $YV$ is the side opposite the $30^{\circ}$ angle and $YV = 24$ units. We use the cosine function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. In right - triangle $YVX$, $\cos30^{\circ}=\frac{YV}{YX}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $YX=\frac{YV}{\cos30^{\circ}}$.

Step3: Calculate the length of YX

Substitute $YV = 24$ into the formula $YX=\frac{YV}{\cos30^{\circ}}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, then $YX=\frac{24}{\frac{\sqrt{3}}{2}}=16\sqrt{3}$ (This is wrong. Let's use the correct 30 - 60 - 90 ratio). In a 30 - 60 - 90 triangle, if the side opposite the $30^{\circ}$ angle is $a$ and the hypotenuse is $c$, $c = 2a$ when considering the relationship between the side opposite the $30^{\circ}$ angle and the hypotenuse. In right - triangle $YVX$, the side opposite the $30^{\circ}$ angle is $YV$. Let $YV=a = 24$ units. The hypotenuse $YX$ (opposite the right - angle) and the correct relationship for a 30 - 60 - 90 triangle gives us: If the side opposite the $30^{\circ}$ angle $YV$ is $a$ and the side opposite the $60^{\circ}$ angle is $b$ and hypotenuse is $c$, and we know that if $a$ is given, and we want to find the side opposite the $60^{\circ}$ angle. The ratio of the sides of a 30 - 60 - 90 triangle is $1:\sqrt{3}:2$. Here, $YV$ is the side opposite the $30^{\circ}$ angle. We want to find the side opposite the $60^{\circ}$ angle $YX$. Since the ratio of the side opposite the $30^{\circ}$ angle to the side opposite the $60^{\circ}$ angle is $1:\sqrt{3}$, if the side opposite the $30^{\circ}$ angle $YV = 12$ (we made a wrong start above, in a 30 - 60 - 90 triangle, if the hypotenuse is considered in terms of the side opposite $30^{\circ}$, but we want the side opposite $60^{\circ}$), and the side opposite the $30^{\circ}$ angle $YV$ is given as 24. The side opposite the $60^{\circ}$ angle $YX$ is $YV\sqrt{3}$. So $YX = 24\sqrt{3}\div2=12\sqrt{3}$ units.

Answer:

$12\sqrt{3}$ units