Question

if these two shapes are similar, what is the measure of the missing length c?
42 mi
28 mi
8 mi
c
c = \square miles

Explanation:

Step1: Set up proportion for similar triangles

Since the triangles are similar, the ratios of corresponding sides are equal. So, $\frac{8}{42}=\frac{c}{28}$.

Step2: Cross - multiply to solve for c

Cross - multiplying gives $42c = 8\times28$. Calculate $8\times28 = 224$. Then, $c=\frac{224}{42}$. Simplify the fraction: divide numerator and denominator by 14, $\frac{224\div14}{42\div14}=\frac{16}{3}\approx5.33$? Wait, no, wait. Wait, maybe I mixed up the corresponding sides. Let's re - check. The side of length 8 in the small triangle corresponds to the side of length 42 in the large triangle? Wait, no, maybe the side of length 8 corresponds to the side of length 28? Wait, no, the small triangle has sides 8 and c, the large has 42 and 28. So the correct proportion is $\frac{8}{x}=\frac{42}{28}$? No, wait, similar triangles: corresponding sides are proportional. Let's label the small triangle sides: 8 and c. Large triangle sides: 42 and 28. So the ratio of the first side of small to first side of large should equal ratio of second side of small to second side of large. So $\frac{8}{42}=\frac{c}{28}$? Wait, no, maybe the 8 corresponds to 28? Wait, no, the small triangle's side 8 and the large triangle's side 28? Wait, no, the large triangle has sides 42 and 28, small has 8 and c. Let's see, maybe the ratio is $\frac{8}{28}=\frac{c}{42}$? Wait, that makes more sense. Let's check: if the small triangle's side 8 corresponds to large triangle's side 28, and small's c corresponds to large's 42. Then the proportion is $\frac{8}{28}=\frac{c}{42}$. Cross - multiply: $28c = 8\times42$. $8\times42 = 336$. Then $c=\frac{336}{28}=12$. Ah, that's better. I had the correspondence wrong. So the correct proportion is $\frac{8}{28}=\frac{c}{42}$ (since 8 and 28 are corresponding sides, c and 42 are corresponding sides). So step1: Identify corresponding sides. Small triangle: 8, c. Large triangle: 28, 42. So proportion $\frac{8}{28}=\frac{c}{42}$. Step2: Cross - multiply: $28c = 8\times42$. Step3: Calculate $8\times42 = 336$. Step4: Divide both sides by 28: $c=\frac{336}{28}=12$.

Answer:

12