Question

find the perimeter of \\( \triangle x y z \\). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since the angles of \(\triangle WUV\) and \(\triangle ZXY\) are \(60^{\circ},63^{\circ},57^{\circ}\) (sum of angles in a triangle is \(180^{\circ}\)), the two triangles are similar by AA (angle - angle) similarity criterion.

Step2: Find the scale factor

We can use the ratio of corresponding sides. Let's take the sides \(WU = 46\) and \(XY=36\). Also, \(WV = 43\) and \(ZY = 34.4\). The scale factor \(k\) from \(\triangle WUV\) to \(\triangle ZXY\) can be found using \(\frac{ZY}{WV}=\frac{34.4}{43}=0.8\)

Step3: Find the lengths of other sides of \(\triangle ZXY\)

The side \(XZ\): Corresponding to \(WU = 46\), so \(XZ=46\times0.8 = 36.8\)
The side \(XY = 36\) (given)
The side \(ZY=34.4\) (given)

Step4: Calculate the perimeter of \(\triangle ZXY\)

Perimeter \(P=XZ + XY+ZY\)
\(P=36.8+36 + 34.4\)
\(P=107.2\)

Answer:

\(107.2\)