Question

1. in the accompanying diagram, δqrs is similar to δlmn, rs = 30, qs = 21, sr = 27, and ln = 7. what is the perimeter of δlmn?

Explanation:

Step1: Find the scale factor

First, identify the corresponding sides. In similar triangles, the ratio of corresponding sides is the scale factor. We know \( QS = 21 \) (wait, no, the problem says \( QS = 21 \)? Wait, the diagram: \( \triangle QRS \) has sides \( QS = 21 \), \( SR = 27 \), \( RQ = 30 \)? Wait, no, the problem says \( RS = 30 \), \( QS = 21 \), \( SR = 27 \)? Wait, maybe typo, probably \( QS = 21 \), \( SR = 27 \), \( RQ = 30 \)? Wait, no, the triangle \( \triangle QRS \) has sides: let's check the diagram. The top triangle: \( Q \), \( R \), \( S \), with \( QR = 30 \), \( QS = 21 \), \( RS = 27 \). Then \( \triangle LMN \) has \( LN = 7 \). So corresponding sides: \( QS \) corresponds to \( LN \)? Wait, \( QS = 21 \), \( LN = 7 \). So scale factor \( k=\frac{LN}{QS}=\frac{7}{21}=\frac{1}{3} \).

Step2: Calculate the perimeter of \( \triangle QRS \)

Perimeter of \( \triangle QRS \) is \( 21 + 27 + 30 = 78 \).

Step3: Find the perimeter of \( \triangle LMN \)

Since the scale factor is \( \frac{1}{3} \), the perimeter of \( \triangle LMN \) is perimeter of \( \triangle QRS \times \) scale factor. So \( 78\times\frac{1}{3}=26 \). Wait, no, wait: maybe the corresponding side is \( QS \) and \( LN \). Wait, \( QS = 21 \), \( LN = 7 \), so scale factor from \( \triangle QRS \) to \( \triangle LMN \) is \( \frac{7}{21}=\frac{1}{3} \). Then perimeter of \( \triangle QRS \) is \( 21 + 27 + 30 = 78 \). Then perimeter of \( \triangle LMN \) is \( 78\times\frac{1}{3}=26 \). Wait, but let's check again. Wait, maybe the sides are \( QS = 21 \), \( SR = 27 \), \( RQ = 30 \). Then perimeter of \( \triangle QRS \) is \( 21 + 27 + 30 = 78 \). Then \( LN = 7 \) corresponds to \( QS = 21 \), so scale factor \( \frac{7}{21}=\frac{1}{3} \). Then perimeter of \( \triangle LMN \) is \( 78\times\frac{1}{3}=26 \).

Answer:

The perimeter of \( \triangle LMN \) is \( \boxed{26} \).