Question

if ( hi = 11 ), ( ij = 10 ), ( hj = 13 ), ( lm = 12 ), and ( km = 15.6 ), find the perimeter of ( \triangle klm ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Find the measure of angle \( J \) in \( \triangle HIJ \)

In a triangle, the sum of angles is \( 180^{\circ} \). So, \( \angle J=180^{\circ}-(50^{\circ}+75^{\circ}) = 55^{\circ} \).

Step2: Establish similarity between \( \triangle HIJ \) and \( \triangle KLM \)

Since \( \angle H = \angle K = 50^{\circ} \) and \( \angle J=\angle M = 55^{\circ} \), by the AA (Angle - Angle) similarity criterion, \( \triangle HIJ\sim\triangle KLM \).

Step3: Find the ratio of similarity

The ratio of similarity \( r=\frac{HI}{KL}\). First, find the perimeter of \( \triangle HIJ\). \(P_{HIJ}=11 + 10+13=34\).
Since \( \triangle HIJ\sim\triangle KLM \), the ratio of their perimeters is the same as the ratio of their corresponding sides. Let the perimeter of \( \triangle KLM\) be \(P_{KLM}\).
We know that the ratio of corresponding sides (using \(HI = 11\) and \(KM = 15.6\), but we should use the correct corresponding sides. Let's use the side - angle - side correspondence. Since \( \angle H\) corresponds to \( \angle K\), \( \angle I\) corresponds to \( \angle L\), \( \angle J\) corresponds to \( \angle M\). The ratio of similarity \(r=\frac{HI}{KL}=\frac{IJ}{LM}=\frac{HJ}{KM}\).
\(r=\frac{HI}{KL}\), but we can also use the perimeter ratio. Let \(P_{KLM}\) be the perimeter of \( \triangle KLM\).
Since the ratio of similarity is \(r\), and \(P_{HIJ}=34\), and using the side - side ratio: \(\frac{HI}{KL}=\frac{IJ}{LM}=\frac{HJ}{KM}\). Let's use \(\frac{IJ}{LM}=\frac{10}{12}=\frac{5}{6}\).
Let \(P_{HIJ}\) and \(P_{KLM}\) be the perimeters of \( \triangle HIJ\) and \( \triangle KLM\) respectively. Since the ratio of perimeters of similar triangles is equal to the ratio of their corresponding sides.
Let \(P_{KLM}=x\). Then \(\frac{P_{HIJ}}{P_{KLM}}=\frac{IJ}{LM}\) (because of similar - triangle property: \( \frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\frac{a + c+e}{b + d + f}\)).
\(\frac{34}{x}=\frac{10}{12}\)
Cross - multiply: \(10x=34\times12\)
\(x=\frac{34\times12}{10}=40.8\)

Answer:

\(40.8\)