Question

find the perimeter of \\( \triangle k l m \\). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since two angles of \(\triangle JHI\) (\(62^{\circ}\) and \(50^{\circ}\)) and two angles of \(\triangle LKM\) (\(62^{\circ}\) and \(68^{\circ}\), as \(180-(62 + 68)=50^{\circ}\)) are equal, \(\triangle JHI\sim\triangle LKM\) by the AA (Angle - Angle) similarity criterion.

Step2: Set up proportion for corresponding sides

The ratio of corresponding sides of similar triangles is equal. Let's find the ratio of sides. For example, if we consider the sides opposite the \(62^{\circ}\) angles: \(\frac{JI}{LM}=\frac{9}{11.7}=\frac{90}{117}=\frac{10}{13}\).
If \(JI = 9\), \(HI=10\), \(JH = 11\) in \(\triangle JHI\) and \(LM = 11.7\), \(LK = 13\), \(MK=x\) in \(\triangle LKM\). Using the ratio \(\frac{HI}{MK}=\frac{10}{x}=\frac{9}{11.7}\) (or using \(\frac{JH}{LK}=\frac{11}{13}\)).
We can also use the ratio \(\frac{JI}{LM}=\frac{HI}{MK}=\frac{JH}{LK}\). Since \(\frac{9}{11.7}=\frac{10}{x}\), cross - multiply: \(9x=10\times11.7\), so \(x = 13\).

Step3: Calculate the perimeter of \(\triangle LKM\)

The perimeter \(P\) of \(\triangle LKM\) is \(P=11.7 + 13+13\).
\(P=37.7\)

Answer:

\(37.7\)