Question

question
if ( u v = 72 ), ( v t = 82 ), and ( y w = 49 ), find the length of ( overline { x y } ). round your answer to the
nearest tenth if necessary. figures are not necessarily drawn to scale.
watch video
show examples

Explanation:

Step1: Find the third angle of each triangle

In \(\triangle TUV\), using the angle - sum property of a triangle (\(180^{\circ}\)), the third angle \(\angle U=180^{\circ}-(59^{\circ} + 44^{\circ})=77^{\circ}\).
In \(\triangle XYW\), the third angle \(\angle W = 180^{\circ}-(44^{\circ}+77^{\circ}) = 59^{\circ}\).
Since \(\angle T=\angle W = 59^{\circ}\), \(\angle V=\angle X = 44^{\circ}\), and \(\angle U=\angle Y = 77^{\circ}\), \(\triangle TUV\sim\triangle XYW\) (by AAA similarity criterion).

Step2: Set up the proportion using the similarity of triangles

For similar triangles \(\triangle TUV\) and \(\triangle XYW\), the ratios of corresponding sides are equal.
We know that \(\frac{UV}{YW}=\frac{VT}{XY}\).
Given \(UV = 72\), \(VT = 82\), and \(YW = 49\).
Substitute the values into the proportion: \(\frac{72}{49}=\frac{82}{XY}\).
Cross - multiply: \(72\times XY=49\times82\).
So, \(XY=\frac{49\times82}{72}\).

Step3: Calculate the value of \(XY\)

First, calculate \(49\times82=(50 - 1)\times82=50\times82-82=4100 - 82 = 4018\).
Then, \(XY=\frac{4018}{72}\approx55.8\).

Answer:

\(55.8\)