Question

find uv.
u t
96
60°
v
write your answer in simplified, rationalized form. do
uv =

Explanation:

Step1: Identify triangle type

Triangle \( UVT \) is right - angled at \( U \), with \( \angle V = 60^{\circ} \), so \( \angle T=30^{\circ} \). In a 30 - 60 - 90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \) (opposite to \( 30^{\circ},60^{\circ},90^{\circ} \) respectively). The side \( UT = 96 \) is opposite to \( \angle V = 60^{\circ} \), and \( UV \) is opposite to \( \angle T = 30^{\circ} \). Let \( UV=x \), \( UT = x\sqrt{3} \).

Step2: Solve for \( UV \)

We know that \( UT = 96=x\sqrt{3} \). To find \( x \) (which is \( UV \)), we solve for \( x \): \( x=\frac{96}{\sqrt{3}} \). Rationalize the denominator by multiplying numerator and denominator by \( \sqrt{3} \): \( x = \frac{96\sqrt{3}}{3}=32\sqrt{3} \). Wait, no, wait. Wait, in a 30 - 60 - 90 triangle, 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 \( 2a \). Here, \( \angle T = 30^{\circ} \), so \( UV \) is opposite \( \angle T \) (30°), \( UT \) is opposite \( \angle V \) (60°). So \( UT = UV\times\sqrt{3} \). Let \( UV = x \), then \( 96=x\sqrt{3}\), so \( x=\frac{96}{\sqrt{3}}=\frac{96\sqrt{3}}{3} = 32\sqrt{3} \)? Wait, no, wait, maybe I mixed up. Wait, angle at \( V \) is 60°, angle at \( U \) is 90°, so angle at \( T \) is 30°. So side \( UV \) is adjacent to 60° and opposite to 30°, side \( UT \) is adjacent to 30° and opposite to 60°, hypotenuse \( VT \). So \( \tan(60^{\circ})=\frac{UT}{UV} \). Since \( \tan(60^{\circ})=\sqrt{3}=\frac{96}{UV} \), so \( UV=\frac{96}{\sqrt{3}} \). Rationalizing, \( UV = \frac{96\sqrt{3}}{3}=32\sqrt{3} \)? Wait, no, \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \), \( \theta = 60^{\circ} \), opposite is \( UT = 96 \), adjacent is \( UV \). So \( \tan(60^{\circ})=\frac{UT}{UV}\Rightarrow\sqrt{3}=\frac{96}{UV}\Rightarrow UV=\frac{96}{\sqrt{3}} = 32\sqrt{3} \). Wait, but also, in 30 - 60 - 90 triangle, the side opposite 30° is half the hypotenuse. Wait, maybe another approach. Let's use trigonometric ratios. \( \tan(\angle V)=\frac{UT}{UV} \). \( \angle V = 60^{\circ} \), \( UT = 96 \), so \( \tan(60^{\circ})=\sqrt{3}=\frac{96}{UV} \), so \( UV=\frac{96}{\sqrt{3}}=\frac{96\sqrt{3}}{3}=32\sqrt{3} \). Wait, no, that can't be. Wait, no, I think I made a mistake. Wait, angle at \( V \) is 60°, so \( \tan(60^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{UT}{UV} \). So \( \sqrt{3}=\frac{96}{UV}\), so \( UV=\frac{96}{\sqrt{3}} = 32\sqrt{3} \). Wait, but let's check with another ratio. \( \cot(60^{\circ})=\frac{UV}{UT} \), and \( \cot(60^{\circ})=\frac{1}{\sqrt{3}} \), so \( \frac{UV}{96}=\frac{1}{\sqrt{3}}\Rightarrow UV=\frac{96}{\sqrt{3}} = 32\sqrt{3} \). Yes, that's correct.

Answer:

\( 32\sqrt{3} \)