Question

these two shapes are similar. find q.
7 cm
21 cm
2 cm
q
q = □ centimeters

Explanation:

Step1: Set up proportion for similar rectangles

Since the rectangles are similar, the ratios of corresponding sides are equal. So, $\frac{7}{q}=\frac{21}{2}$ (or $\frac{21}{7}=\frac{q}{2}$, either way works as long as corresponding sides are proportional). Let's use $\frac{7}{q}=\frac{21}{2}$.

Step2: Cross - multiply to solve for q

Cross - multiplying gives $21\times q = 7\times2$. So, $21q = 14$. Then, divide both sides by 21: $q=\frac{14}{21}$. Simplify the fraction: $\frac{14\div7}{21\div7}=\frac{2}{3}$? Wait, no, wait. Wait, maybe I mixed up the corresponding sides. Wait, the first rectangle has length 21 and height 7, the second has height 2 and length q? Wait, no, maybe the 7 corresponds to 2, and 21 corresponds to q? Let's re - check.

Wait, similar figures: corresponding sides are proportional. So if the first rectangle has dimensions 21 cm (length) and 7 cm (width), and the second has length q and width 2 cm. Then the ratio of length to width should be the same. So $\frac{21}{7}=\frac{q}{2}$.

Step3: Solve the correct proportion

$\frac{21}{7}=\frac{q}{2}$. Simplify $\frac{21}{7}=3$. So $3=\frac{q}{2}$. Multiply both sides by 2: $q = 3\times2=6$? Wait, no, wait, that can't be. Wait, no, maybe the 7 corresponds to q and 21 corresponds to 2? No, that would be wrong. Wait, let's look at the figures. The first rectangle is longer, the second is shorter. So the scale factor: the smaller rectangle's side of 2 cm corresponds to the larger rectangle's side of 7 cm? No, that doesn't make sense. Wait, no, the first rectangle has length 21 and height 7, the second has height 2 and length q. So the ratio of height to length for the first is $\frac{7}{21}=\frac{1}{3}$. So for the second, $\frac{2}{q}=\frac{1}{3}$. Then cross - multiply: $q = 2\times3 = 6$? Wait, no, $\frac{7}{21}=\frac{1}{3}$, so the height of the small one is 2, so the length should be $2\div\frac{1}{3}=6$? Wait, no, maybe I had the ratio reversed. Let's do it properly.

Let the two rectangles be Rectangle A (21 cm by 7 cm) and Rectangle B (q cm by 2 cm). Since they are similar, $\frac{\text{length of A}}{\text{length of B}}=\frac{\text{width of A}}{\text{width of B}}$. So $\frac{21}{q}=\frac{7}{2}$. Now cross - multiply: $7\times q=21\times2$. So $7q = 42$. Then divide both sides by 7: $q=\frac{42}{7}=6$. Yes, that's correct.

Answer:

6