Question

if these two shapes are similar, what is the measure of the missing length c? 10 cm 5 cm 2 cm c = centimeters

Explanation:

Step1: Set up proportion for similar shapes

Since the shapes are similar, the ratios of corresponding sides are equal. Let the sides of the larger shape be 10 cm and 5 cm, and the corresponding sides of the smaller shape be \( c \) and 2 cm. So we set up the proportion \(\frac{10}{5}=\frac{c}{2}\) (or we can also use \(\frac{5}{10}=\frac{2}{c}\), but the first one is more intuitive as 10 corresponds to \( c \) and 5 corresponds to 2, depending on the orientation of the similar parallelograms). Wait, actually, let's check the corresponding sides. The larger parallelogram has sides 10 cm and 5 cm, the smaller has sides \( c \) and 2 cm. So the ratio of the larger side to the smaller corresponding side should be equal. So \(\frac{10}{c}=\frac{5}{2}\) (cross - multiplying: \(5c = 10\times2\)).

Step2: Solve for \( c \)

From the proportion \(\frac{10}{c}=\frac{5}{2}\), cross - multiply to get \(5c=10\times2\). Then \(5c = 20\). Divide both sides by 5: \(c=\frac{20}{5}=4\)? Wait, no, wait. Wait, maybe I mixed up the corresponding sides. Let's re - examine. The larger parallelogram: one side is 5 cm, the other is 10 cm. The smaller parallelogram: one side is 2 cm, the other is \( c \). Since they are similar, the ratio of the sides of the larger to the smaller should be the same. So \(\frac{5}{2}=\frac{10}{c}\)? No, wait, maybe the 5 cm side of the larger corresponds to the 2 cm side of the smaller, and the 10 cm side of the larger corresponds to the \( c \) side of the smaller. So the ratio of similarity is \(\frac{5}{2}\) (scale factor from smaller to larger) or \(\frac{2}{5}\) (scale factor from larger to smaller). Wait, let's do it correctly. For similar figures, the ratio of corresponding sides is equal. Let's assume that the side of length 5 cm in the larger shape corresponds to the side of length 2 cm in the smaller shape, and the side of length 10 cm in the larger shape corresponds to the side of length \( c \) in the smaller shape. So the proportion is \(\frac{5}{2}=\frac{10}{c}\). Cross - multiplying: \(5c=10\times2\), \(5c = 20\), \(c = 4\)? Wait, no, that would make the smaller shape larger, which is wrong. Wait, maybe the 10 cm side corresponds to the 2 cm side? No, that can't be. Wait, maybe the correct proportion is \(\frac{10}{5}=\frac{c}{2}\). Let's check: \(\frac{10}{5}=2\), so \(\frac{c}{2}=2\), then \(c = 4\)? Wait, no, that would mean the smaller shape has a side of 4, but the other side is 2. Wait, the larger shape has sides 10 and 5 (ratio 2:1), the smaller should also have a ratio of 2:1. So if one side is 2, the other should be 4? Wait, but 10 and 5: 10/5 = 2. So 2 times 2 is 4? Wait, maybe I had the proportion reversed. Let's think again. The larger parallelogram: sides 5 and 10 (so the ratio of the two sides is 5:10 = 1:2). The smaller parallelogram: sides 2 and \( c \), so the ratio should also be 1:2. So \(\frac{2}{c}=\frac{1}{2}\), then \(c = 4\)? Wait, no, \(\frac{2}{c}=\frac{1}{2}\) gives \(c = 4\). Alternatively, if the ratio of larger to smaller is 5:2 (for the corresponding sides), then 10 divided by (5/2) is 10*(2/5)=4. Yes, that makes sense. So the missing length \( c \) is 4.

Answer:

4