Question

triangle ijk is similar to triangle lmn. find the measure of side lm. round your answer to the nearest tenth if necessary. figures are not drawn to scale.
k
59
i 39 j
n 11 l
m

Explanation:

Step1: Identify corresponding sides

Since \(\triangle IJK \sim \triangle LMN\), the corresponding sides are proportional. Let's assume \(IJ\) corresponds to \(LN\), \(IK\) corresponds to \(LM\), and \(JK\) corresponds to \(MN\) (or vice - versa based on the right - angled nature, assuming \(\angle I=\angle L = 90^{\circ}\)). So we have the proportion \(\frac{IK}{LM}=\frac{IJ}{LN}\). Wait, maybe a better way: Let's check the sides. From the diagram, \(IK = 59\), \(IJ=39\), \(LN = 11\). Let's assume the correspondence is \(I ightarrow L\), \(J ightarrow N\), \(K ightarrow M\). So the proportion is \(\frac{IK}{LM}=\frac{IJ}{LN}\). Wait, no, maybe \(IJ\) and \(LN\) are corresponding, \(IK\) and \(LM\) are corresponding. Wait, let's re - establish the similarity. If \(\triangle IJK\sim\triangle LMN\), then \(\frac{IJ}{LM}=\frac{IK}{LN}\)? No, maybe I got the correspondence wrong. Wait, the right angle: \(\angle I\) and \(\angle L\) are right angles (since \(IK\) is vertical and \(IJ\) is horizontal, \(LN\) is horizontal and \(LM\) is vertical). So \(\angle I=\angle L = 90^{\circ}\), \(\angle J=\angle N\), \(\angle K=\angle M\). So the sides: \(IJ\) (horizontal in \(\triangle IJK\)) corresponds to \(LN\) (horizontal in \(\triangle LMN\)), \(IK\) (vertical in \(\triangle IJK\)) corresponds to \(LM\) (vertical in \(\triangle LMN\)), and \(JK\) corresponds to \(MN\). So the proportion is \(\frac{IK}{LM}=\frac{IJ}{LN}\). Wait, \(IK = 59\), \(IJ = 39\), \(LN=11\). We need to find \(LM\). So cross - multiplying, \(LM=\frac{IK\times LN}{IJ}\). Wait, no, if \(\frac{IK}{LM}=\frac{IJ}{LN}\), then \(LM=\frac{IK\times LN}{IJ}\)? Wait, no, let's write the proportion correctly. If \(\triangle IJK\sim\triangle LMN\), then \(\frac{IK}{LM}=\frac{IJ}{LN}\) (because \(IK\) and \(LM\) are the vertical sides, \(IJ\) and \(LN\) are the horizontal sides). So \(LM=\frac{IK\times LN}{IJ}\)? Wait, \(IK = 59\), \(IJ = 39\), \(LN = 11\). So \(LM=\frac{59\times11}{39}\)? Wait, no, maybe the correspondence is \(\frac{IJ}{LN}=\frac{IK}{LM}\). So \(LM=\frac{IK\times LN}{IJ}\). Let's calculate that.

Step2: Calculate \(LM\)

\(IK = 59\), \(LN=11\), \(IJ = 39\). So \(LM=\frac{59\times11}{39}=\frac{649}{39}\approx16.6\) (rounded to the nearest tenth). Wait, maybe I had the correspondence wrong. Wait, maybe \(IJ\) corresponds to \(LM\) and \(IK\) corresponds to \(LN\). Let's try that. If \(\frac{IJ}{LM}=\frac{IK}{LN}\), then \(LM=\frac{IJ\times LN}{IK}\). Then \(LM=\frac{39\times11}{59}=\frac{429}{59}\approx7.3\). That doesn't seem right. Wait, maybe the right - angled sides: in \(\triangle IJK\), the legs are \(IJ = 39\) and \(IK = 59\). In \(\triangle LMN\), the legs are \(LN = 11\) and \(LM\) (the one we need to find). Since the triangles are similar, the ratio of corresponding legs should be equal. So if \(\triangle IJK\sim\triangle LMN\), then \(\frac{IJ}{LN}=\frac{IK}{LM}\) (if \(IJ\) corresponds to \(LN\) and \(IK\) corresponds to \(LM\)). Then \(LM=\frac{IK\times LN}{IJ}=\frac{59\times11}{39}=\frac{649}{39}\approx16.6\). Or if \(\frac{IJ}{LM}=\frac{IK}{LN}\), then \(LM=\frac{IJ\times LN}{IK}=\frac{39\times11}{59}\approx7.3\). Wait, maybe the correspondence is \(I ightarrow N\), \(J ightarrow L\), \(K ightarrow M\). Then the legs: \(IJ\) corresponds to \(NL\), \(IK\) corresponds to \(NM\), \(JK\) corresponds to \(LM\). No, this is confusing. Wait, let's look at the diagram again. The first triangle has a vertical side of length 59 (IK) and a horizontal side of length 39 (IJ). The second triangle has a horizontal side of length 11 (LN) and a vertical side of length \(LM\) (which we need to find). Since they are similar, the ratio of vertical to horizontal should be the same. So for the first triangle, the ratio of vertical (IK = 59) to horizontal (IJ = 39) is \(\frac{59}{39}\). For the second triangle, the ratio of vertical (LM) to horizontal (LN = 11) should be the same. So \(\frac{LM}{11}=\frac{59}{39}\), so \(LM=\frac{59\times11}{39}=\frac{649}{39}\approx16.6\).

Answer:

\(16.6\)