Question

question
quadrilateral ijkl is similar to quadrilateral mnop. find the measure of side pm. round your answer to the nearest tenth if necessary.
(images of quadrilaterals ijkl and mnop: ijkl has sides l to i = 8, j to k = 5; mnop has side n to o = 19.7)

Explanation:

Step1: Identify Corresponding Sides

Since quadrilaterals \(IJKL\) and \(MNOP\) are similar, their corresponding sides are proportional. Side \(JK = 5\) in \(IJKL\) corresponds to side \(NO = 19.7\) in \(MNOP\), and side \(LI = 8\) in \(IJKL\) corresponds to side \(PM\) in \(MNOP\).

Step2: Set Up Proportion

Let \(PM = x\). The proportion of corresponding sides is \(\frac{JK}{NO}=\frac{LI}{PM}\), so \(\frac{5}{19.7}=\frac{8}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(5x = 8\times19.7\). Calculate \(8\times19.7 = 157.6\). Then \(x=\frac{157.6}{5}=31.52\). Rounding to the nearest tenth (if necessary, but here it's already precise enough in tenths as \(31.52\approx31.5\)? Wait, no, wait: \(157.6\div5 = 31.52\), which to the nearest tenth is \(31.5\)? Wait, no, \(31.52\) to the nearest tenth: the hundredth digit is 2, which is less than 5, so we keep the tenth digit as 5. Wait, but let's check the proportion again. Wait, maybe the correspondence is different. Wait, maybe \(JK\) (length 5) corresponds to \(NO\) (length 19.7), and \(LI\) (length 8) corresponds to \(PM\). So the ratio of similarity is \(\frac{NO}{JK}=\frac{19.7}{5}\). Then \(PM = LI\times\frac{NO}{JK}=8\times\frac{19.7}{5}\). Let's calculate that: \(\frac{19.7}{5}=3.94\), then \(8\times3.94 = 31.52\), which is \(31.5\) when rounded to the nearest tenth? Wait, no, \(31.52\) to the nearest tenth is \(31.5\)? Wait, no, \(31.52\): the tenths place is 5, hundredths is 2. So we round down, so \(31.5\)? Wait, no, \(31.52\) is \(31.5\) when rounded to the nearest tenth? Wait, no, \(31.52\) is closer to \(31.5\) than \(31.6\) because the hundredth digit is 2. But let's check the calculation again. \(8\times19.7 = 157.6\), \(157.6\div5 = 31.52\). So the measure of \(PM\) is \(31.5\) (rounded to the nearest tenth) or \(31.52\). But let's see, maybe the problem expects the exact value or the rounded value. Wait, the problem says "round your answer to the nearest tenth if necessary". So \(31.52\) rounded to the nearest tenth is \(31.5\)? Wait, no, \(31.52\) to the nearest tenth: the number is \(31.5\) (tenths place) and \(2\) (hundredths). Since \(2 < 5\), we round down, so \(31.5\). But wait, maybe I had the correspondence wrong. Wait, maybe \(JK\) is vertical, \(NO\) is vertical, \(LI\) is left - side, \(PM\) is left - side. So the ratio of similarity is \(\frac{NO}{JK}=\frac{19.7}{5}\), so the length of \(PM\) is \(LI\times\frac{NO}{JK}=8\times\frac{19.7}{5}=31.52\approx31.5\) (to the nearest tenth) or \(31.52\) as is. But let's check the calculation again. \(19.7\div5 = 3.94\), \(8\times3.94 = 31.52\). So the answer is \(31.5\) (rounded to the nearest tenth) or \(31.52\). But maybe the problem doesn't require rounding? Wait, the question says "round your answer to the nearest tenth if necessary". Let's see, \(31.52\) to the nearest tenth is \(31.5\).

Answer:

\(31.5\) (or \(31.52\) if we don't round, but following the "nearest tenth" instruction, it's \(31.5\))