Question

find the perimeter or the area.

1. $\parallelogram abcd \sim \parallelogram efgh$. the ratio of a pair of corresponding sides is 2:3. the perimeter of $\parallelogram abcd$ is 24 cm. find the perimeter of $\parallelogram efgh$.
2. $\triangle jkl \sim \triangle mnp$. the ratio of a pair of corresponding sides is 1 to 2. the area of $\triangle jkl$ is $10\\ \mathrm{cm}^2$. find the area of $\triangle mnp$.

problem solving
solve each problem. show your work.

1. a 6-ft-tall man casts a 2-ft shadow. at the same time, a tree casts a 3-ft shadow. how tall is the tree?
2. the scale on a scale drawing is $\frac{1}{4}$ in. $= 1$ ft. a room on the drawing is $1\frac{1}{2}$ in. long. what is the actual length of the room?

Explanation:

Problem 9

Step1: Recall similarity of parallelograms

For similar figures, the ratio of perimeters is equal to the ratio of corresponding sides. Let the perimeter of \(ABCD\) be \(P_1 = 24\) cm, perimeter of \(EFGH\) be \(P_2\), and the ratio of corresponding sides be \(2:3\) (so ratio of perimeters is also \(2:3\)).

Step2: Set up proportion

\(\frac{P_1}{P_2}=\frac{2}{3}\)
Substitute \(P_1 = 24\) into the proportion: \(\frac{24}{P_2}=\frac{2}{3}\)
Cross - multiply: \(2\times P_2=24\times3\)
\(2P_2 = 72\)
Divide both sides by 2: \(P_2=\frac{72}{2}=36\)

Step1: Recall area ratio of similar triangles

For similar triangles, the ratio of areas is equal to the square of the ratio of corresponding sides. The ratio of corresponding sides is \(1:2\), so the ratio of areas is \((\frac{1}{2})^2=\frac{1}{4}\)
Let the area of \(\triangle JKL\) be \(A_1 = 10\) \(cm^2\) and area of \(\triangle MNP\) be \(A_2\)

Step2: Set up proportion

\(\frac{A_1}{A_2}=\frac{1}{4}\)
Substitute \(A_1 = 10\) into the proportion: \(\frac{10}{A_2}=\frac{1}{4}\)
Cross - multiply: \(A_2=10\times4 = 40\)

Step1: Set up proportion (similar triangles)

At the same time, the triangles formed by the man, his shadow and the tree, its shadow are similar. Let the height of the man be \(h_1 = 6\) ft, length of his shadow \(s_1 = 2\) ft, length of tree's shadow \(s_2 = 3\) ft and height of tree be \(h_2\)
The proportion is \(\frac{h_1}{s_1}=\frac{h_2}{s_2}\)

Step2: Solve for \(h_2\)

Substitute \(h_1 = 6\), \(s_1 = 2\), \(s_2 = 3\) into the proportion: \(\frac{6}{2}=\frac{h_2}{3}\)
Cross - multiply: \(2\times h_2=6\times3\)
\(2h_2 = 18\)
Divide both sides by 2: \(h_2=\frac{18}{2}=9\)

Answer:

36 cm

Problem 10