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 the right - triangle relationship

In right - triangle YVX, YV is the side opposite the 30 - degree angle and YX is the hypotenuse. The ratio of the sides in a 30 - 60 - 90 triangle is $1:\sqrt{3}:2$, where the side opposite the 30 - degree angle is $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

Step2: Apply the 30 - 60 - 90 triangle ratio

We know that the side opposite the 30 - degree angle (YV) has length 24 units. Let the side opposite the 30 - degree angle be $a$. So $a = 24$. The hypotenuse YX (let it be $c$) and in a 30 - 60 - 90 triangle, if the side opposite 30 - degree angle is $a$, the hypotenuse $c = 2a$. But we can also use the cosine function. $\cos(30^{\circ})=\frac{YV}{YX}$. Since $\cos(30^{\circ})=\frac{\sqrt{3}}{2}$ and $YV = 24$, we have $YX=\frac{YV}{\cos(30^{\circ})}=\frac{24}{\frac{\sqrt{3}}{2}}=16\sqrt{3}$ (this is wrong approach above, correct is using the ratio). In a 30 - 60 - 90 triangle, if the side opposite 30 - degree angle is $a$, and here $a = 12$ (because the side opposite 30 - degree angle in right - triangle YVX where hypotenuse is YX and one side is YV), the hypotenuse $YX = 12\sqrt{3}$ (using the ratio of sides $a:a\sqrt{3}:2a$ where the side opposite 30 - degree angle is $a$ and we know from the ratio that if the non - hypotenuse side adjacent to 30 - degree angle is $a$, and the hypotenuse is $2a$ and the other non - hypotenuse side is $a\sqrt{3}$. Here we know the side opposite 30 - degree angle in right - triangle YVX is half of the hypotenuse. Since the side opposite 30 - degree angle (YV) is 12, the hypotenuse $YX = 12\sqrt{3}$).

Answer:

D. $12\sqrt{3}$ units