Question

1. write and solve a proportion to find the distance across the lake. 21) tell whether the two triangles are similar. explain your reasoning. 22) the polygons are similar. find x. 23) draw an example of a convex polygon. 24) draw an example of a concave polygon.

Explanation:

20)

Step1: Set up the proportion

Since the two - triangle situation forms similar triangles, the ratios of corresponding sides are equal. The proportion is $\frac{x}{500}=\frac{400}{100}$.

Step2: Cross - multiply

Cross - multiplying gives us $100x = 400\times500$.

Step3: Solve for x

$100x=200000$, then $x=\frac{200000}{100}=2000$ ft.

Step1: Check the ratios of corresponding sides

For the first triangle with sides 3, 4, 5 and the second triangle with sides 6, 8, 10. Calculate the ratios: $\frac{6}{3}=2$, $\frac{8}{4}=2$, $\frac{10}{5}=2$.

Step2: Apply the similarity criterion

Since the ratios of the corresponding sides of the two triangles are equal, by the Side - Side - Side (SSS) similarity criterion, the two triangles are similar.

Step1: Set up the proportion for similar polygons

For similar polygons, the ratios of corresponding sides are equal. So, $\frac{x}{3}=\frac{10}{4}$.

Step2: Cross - multiply

Cross - multiplying gives $4x=3\times10$.

Step3: Solve for x

$4x = 30$, then $x=\frac{30}{4}=7.5$.

Answer:

$x = 2000$ ft

21)