Question

the perimeter of △mns is 36 units. what is the length of (overline{rt})? a. 12 units b. 20 units c. 24 units d. 28 units

Explanation:

Step1: Find the scale factor

Since the triangles are similar (by the properties of similar triangles, assume \(\triangle LMN\sim\triangle RST\)), the ratio of corresponding sides is the scale factor. The ratio of \(MN\) to \(RS\) is \(\frac{MN}{RS}\). Let's assume the scale - factor \(k\). Given \(MN = 6\) (from the side - length relation in similar triangles, assume \(MN\) corresponds to \(RS\)), and from the sides \(LM = 4x\), \(LN=2x\), \(MN\) (let's first find the perimeter of \(\triangle LMN\)). The perimeter of \(\triangle LMN\) is \(P_{LMN}=4x + 2x+6=6x + 6\). But we know that for similar triangles, the ratio of perimeters is equal to the ratio of corresponding sides.

Another way: The ratio of corresponding sides. Let's assume the ratio of \(LN\) to \(RS\) (corrected approach). The ratio of the sides of \(\triangle LMN\) to \(\triangle RST\): \(\frac{LN}{RS}=\frac{2x}{6}\), \(\frac{LM}{ST}=\frac{4x}{8}\). Since \(\frac{2x}{6}=\frac{4x}{8}\) (for similar triangles, ratios of corresponding sides are equal), cross - multiply \(2x\times8 = 4x\times6\) (which is not correct, better use perimeter ratio).

The correct approach: The ratio of the sides of \(\triangle LMN\) (perimeter \(P_{LMN}=4x + 2x+6=6x + 6\)) and \(\triangle RST\) (perimeter \(P_{RST}=36\)). The ratio of perimeters of similar triangles is equal to the ratio of corresponding sides. Let the ratio of \(\triangle LMN\) to \(\triangle RST\) be \(\frac{2x}{6}\) (assuming \(LN\) corresponds to \(RS\)). The perimeter of \(\triangle LMN\): \(P_{LMN}=4x+2x + 6=6x + 6\). But we know that \(\frac{P_{LMN}}{P_{RST}}=\frac{LN}{RS}\) (perimeter ratio = side - length ratio for similar triangles). Wait, no, better use the fact that if \(\triangle LMN\sim\triangle RST\), then \(\frac{LM}{ST}=\frac{LN}{RS}=\frac{MN}{RT}\).

Let's use the ratio of sides: \(\frac{LM}{ST}=\frac{4x}{8}\), \(\frac{LN}{RS}\). Since \(\triangle LMN\sim\triangle RST\), the ratio of sides is constant. Let's assume \(LN\) corresponds to \(RS\) (\(LN = 2x\), \(RS = 6\)), \(LM\) corresponds to \(ST\) (\(LM = 4x\), \(ST = 8\)). The ratio \(\frac{LM}{ST}=\frac{4x}{8}=\frac{x}{2}\), \(\frac{LN}{RS}=\frac{2x}{6}=\frac{x}{3}\) (wrong, so re - do).

Correct: The two triangles \(\triangle LMN\) and \(\triangle RST\) are similar. The ratio of their perimeters is equal to the ratio of their corresponding sides. Let the ratio of \(\triangle LMN\) to \(\triangle RST\) be \(k\). The perimeter of \(\triangle LMN\): \(P_{LMN}=4x+2x + 6=6x + 6\). The perimeter of \(\triangle RST\) is \(P_{RST}=36\). Also, \(\frac{LM}{ST}=\frac{4x}{8}\), \(\frac{LN}{RS}\) (assume \(LN\) corresponds to \(RS\)).

Another approach: Since \(\triangle LMN\sim\triangle RST\) (by AA, SSS, or SAS similarity, assume they are similar from the problem structure). The ratio of the sides: \(\frac{LM}{ST}=\frac{LN}{RS}\). Let \(LN = 2x\), \(RS = 6\), \(LM = 4x\), \(ST = 8\). \(\frac{4x}{8}=\frac{2x}{6}\) (wrong). Correct: Use the fact that the ratio of perimeters of similar triangles is equal to the ratio of corresponding sides.

Let the ratio of \(\triangle LMN\) to \(\triangle RST\) be \(r\). The perimeter of \(\triangle LMN\): \(P_{LMN}=4x+2x + 6=6x + 6\). But we know that \(\frac{P_{LMN}}{P_{RST}}=\frac{LN}{RS}\) (perimeter ratio = side ratio). Wait, no, if \(\triangle LMN\sim\triangle RST\), then \(\frac{LN}{RS}=\frac{LM}{ST}=\frac{MN}{RT}\). Also, \(P_{LMN}=4x + 2x+MN\), \(P_{RST}=36\).

Since \(\frac{LM}{ST}=\frac{4x}{8}\), \(\frac{LN}{RS}\) (assume \(RS = 6\)). Let's find \(x\) from the perimeter of \(\triangle LMN\). If \(\triangle LMN\sim\triangle RST\), then \(\frac{LM + LN+MN}{ST + RS+RT}=\frac{LM}{ST}\). Let \(LM = 4x\), \(LN = 2x\), \(MN\) (corresponds to \(RT\)), \(ST = 8\), \(RS = 6\). The ratio \(\frac{4x+2x + MN}{36}=\frac{4x}{8}\). Also, \(\frac{MN}{RT}=\frac{4x}{8}\).

Alternatively, since \(\frac{LM}{ST}=\frac{4x}{8}=\frac{x}{2}\), \(\frac{LN}{RS}=\frac{2x}{6}=\frac{x}{3}\) (wrong, so the triangles are similar with ratio \(\frac{1}{2}\)). Wait, \(\frac{ST}{LM}=\frac{8}{4x}\), \(\frac{RS}{LN}=\frac{6}{2x}\). Since \(\frac{8}{4x}=\frac{6}{2x}\) (cross - multiply \(8\times2x=4x\times6\), \(16x = 24x\) (wrong)).

Correct approach: The perimeter of \(\triangle RST\) is \(36\). Assume \(\triangle LMN\sim\triangle RST\) with \(\frac{LN}{RS}=\frac{LM}{ST}\). Let \(LN = 2x\), \(RS = 6\), \(LM = 4x\), \(ST = 8\). The ratio of perimeters: \(\frac{P_{LMN}}{P_{RST}}=\frac{LN}{RS}\). The perimeter of \(\triangle LMN\): \(P_{LMN}=4x + 2x+6=6x + 6\). \(\frac{6x + 6}{36}=\frac{2x}{6}\) (perimeter ratio = side ratio). Cross - multiply: \((6x + 6)\times6=36\times2x\).

\[ \begin{align*} 36x+36&=72x\\ 36&=72x - 36x\\ 36x&=36\\ x& = 1 \end{align*} \]

Now, since \(\triangle LMN\sim\triangle RST\), \(\frac{LM}{ST}=\frac{LN}{RS}=\frac{MN}{RT}\). \(\frac{LM}{ST}=\frac{4\times1}{8}=\frac{1}{2}\). The perimeter of \(\triangle RST\) is \(36\). Let \(RT\) be the side we want to find. The sum of the known sides of \(\triangle RST\) is \(ST + RS=8 + 6=14\). Let \(RT=y\). Then \(14 + y=36\) (no, wrong, because of similarity).

Since \(\frac{MN}{RT}=\frac{1}{2}\) (from \(\frac{LM}{ST}=\frac{4x}{8}\), \(x = 1\), so \(\frac{4}{8}=\frac{1}{2}\)). The length of \(MN\) (in \(\triangle LMN\), when \(x = 1\), \(MN = 6\)). Then \(\frac{6}{RT}=\frac{1}{2}\), so \(RT = 12\).

Answer:

A. 12 units