Question

a flashlight is projecting a triangle onto a wall, as shown below. the original triangle and its projection are similar. what is the missing length n on the projection? 28 19.2 13.3 30

Explanation:

Step1: Set up proportion for similar triangles

Since the triangles are similar, the ratios of corresponding sides are equal. So, $\frac{16}{20}=\frac{24}{n}$.

Step2: Cross - multiply to solve for n

Cross - multiplying gives $16n = 20\times24$.

Step3: Calculate the right - hand side

$20\times24 = 480$, so the equation becomes $16n=480$.

Step4: Solve for n

Divide both sides by 16: $n=\frac{480}{16}=30$. Wait, no, wait. Wait, maybe I mixed up the corresponding sides. Let's re - check. The original triangle has sides 16, 16, 24. The projection has sides 20, 20, n. So the ratio of original side to projection side should be consistent. So the ratio of 16 (original) to 20 (projection) is the same as 24 (original) to n (projection). Wait, no, maybe the original triangle's base is 24 and the projection's base is n, and the equal sides are 16 (original) and 20 (projection). So the ratio of similarity is $\frac{20}{16}=\frac{5}{4}$. Then the base of the projection $n$ should be $24\times\frac{5}{4}=30$? Wait, but let's do it correctly. For similar triangles, corresponding sides are proportional. So if the original triangle has a side of length 16 and the projection has the corresponding side of length 20, and the original triangle has a base of 24, then the projection's base $n$ is given by $\frac{16}{20}=\frac{24}{n}$? No, that would be if 16 corresponds to 20 and 24 corresponds to n. Wait, cross - multiplying: $16n = 20\times24$, $16n = 480$, $n=\frac{480}{16}=30$. Wait, but let's check the ratio again. The ratio of the sides: 16 to 20 is $\frac{4}{5}$, so the base of the original is 24, so the base of the projection should be $24\div\frac{4}{5}=24\times\frac{5}{4}=30$. Yes, that's correct.

Answer:

30