Question

triangle ghi is similar to triangle jkl. find the measure of side jk. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify corresponding sides

Since \(\triangle GHI \sim \triangle JKL\), the ratio of corresponding sides is equal. Let \(GI = 5\), \(JH = 19\) (wait, no, \(GI\) corresponds to \(JL\), and \(HI\) corresponds to \(JK\)? Wait, looking at the triangles: \(GI = 5\), \(JL = 19\), and \(HI = 7.4\), \(JK\) is what we need. So the ratio of similarity is \(\frac{JL}{GI}=\frac{19}{5}\).

Step2: Set up proportion for \(JK\)

Using the similarity ratio, \(\frac{JK}{HI}=\frac{JL}{GI}\). Substitute the known values: \(\frac{JK}{7.4}=\frac{19}{5}\).

Step3: Solve for \(JK\)

Multiply both sides by \(7.4\): \(JK = \frac{19\times7.4}{5}\). Calculate \(19\times7.4 = 140.6\), then divide by \(5\): \(JK=\frac{140.6}{5}=28.12\), which rounds to \(28.1\) (wait, no, 140.6 divided by 5 is 28.12? Wait 197.4: 207.4=148, minus 17.4=7.4, so 148-7.4=140.6. Then 140.6/5=28.12, which to the nearest tenth is 28.1? Wait, no, 28.12 rounded to the nearest tenth is 28.1? Wait, 28.12: the tenths place is 1, hundredths is 2, so we keep it 28.1? Wait, no, 28.12 is 28.1 when rounded to the nearest tenth? Wait, no, 28.12: the first decimal is 1, second is 2, so since 2 < 5, we don't round up. Wait, but let's check the calculation again. Wait, maybe I mixed up the sides. Wait, \(GI\) is 5, \(JL\) is 19, so the scale factor from \(\triangle GHI\) to \(\triangle JKL\) is \(19/5 = 3.8\). Then \(HI\) is 7.4, so \(JK = HI \times 3.8 = 7.4 \times 3.8\). Let's calculate that: 73.8=26.6, 0.4*3.8=1.52, so total 26.6+1.52=28.12, which is 28.1 when rounded to the nearest tenth. Wait, but maybe the original problem had \(HI = 7.4\) (maybe a typo, 7.4? Or 7.4 is correct). So the calculation is correct.

Answer:

\(28.1\) (Wait, but let's check again. Wait, 7.4 times 3.8: 7.43 = 22.2, 7.40.8=5.92, so 22.2+5.92=28.12, which is 28.1 when rounded to the nearest tenth. So the answer is 28.1.