Question

1. parallelogram abcd is similar to parallelogram efgh. what is the length of gh?
2. the rectangular prisms shown below are similar. what are the values of x and y?
3. the triangular prisms shown below are similar. what are the values of x and y?
4. the figures below are similar. what is the length of x?

Explanation:

Problem 1: Parallelogram Similarity (Length of \( \overline{GH} \))

Step 1: Identify Corresponding Sides

Parallelogram \( ABCD \) is similar to \( EFGH \). Corresponding sides: \( AB = 21 \) in, \( AD = 6 \) in, \( EF = 7 \) in, and \( \overline{GH} \) corresponds to \( AB \)? Wait, no—wait, \( ABCD \): \( AB = 21 \), \( AD = 6 \); \( EFGH \): \( EF = 7 \), \( EH =? \), \( \overline{GH} \) corresponds to \( AB \)? Wait, similarity ratio: \( \frac{EF}{AB} = \frac{7}{21} = \frac{1}{3} \)? No, wait, \( AD \) and \( EH \)? Wait, no, let's check: \( ABCD \) has sides \( AB = 21 \) in, \( AD = 6 \) in; \( EFGH \) has \( EF = 7 \) in, \( EH =? \), \( \overline{GH} \) is parallel to \( EF \)? Wait, no, in parallelograms, opposite sides are equal. So \( AB \parallel CD \), \( AD \parallel BC \); \( EF \parallel GH \), \( EH \parallel FG \). So corresponding sides: \( AB \) corresponds to \( EF \)? No, \( AB \) length 21, \( EF \) length 7. So ratio is \( \frac{EF}{AB} = \frac{7}{21} = \frac{1}{3} \). Then \( \overline{GH} \) corresponds to \( AB \)? Wait, no, \( GH \) is equal to \( EF \)? No, wait, no—wait, \( ABCD \): \( AB = 21 \), \( AD = 6 \); \( EFGH \): \( EF = 7 \), \( EH =? \). Wait, maybe \( AD \) corresponds to \( EH \)? Wait, the problem says "Parallelogram \( ABCD \) is similar to parallelogram \( EFGH \). What is the length of \( \overline{GH} \)?" So \( AB \) corresponds to \( EF \), \( BC \) to \( FG \), \( CD \) to \( GH \), \( DA \) to \( HE \). So \( AB = 21 \), \( EF = 7 \); ratio \( \frac{EF}{AB} = \frac{7}{21} = \frac{1}{3} \). Then \( CD = AB = 21 \)? No, wait, \( CD = AB = 21 \), so \( GH = CD \times \) ratio? Wait, no, similarity ratio: if \( ABCD \sim EFGH \), then \( \frac{EF}{AB} = \frac{FG}{BC} = \frac{GH}{CD} = \frac{HE}{DA} \). So \( AB = 21 \), \( EF = 7 \), so ratio \( k = \frac{7}{21} = \frac{1}{3} \). Then \( GH = CD \times k \). But \( CD = AB = 21 \)? No, \( CD = AB = 21 \), so \( GH = 21 \times \frac{1}{3} = 7 \)? Wait, no, \( EF = 7 \), which is \( AB \times \frac{1}{3} \), so \( GH \) should be \( CD \times \frac{1}{3} \), but \( CD = AB = 21 \), so \( GH = 7 \)? Wait, no, maybe I mixed up. Wait, \( ABCD \): \( AB = 21 \), \( AD = 6 \); \( EFGH \): \( EF = 7 \), \( EH =? \). Wait, \( AD = 6 \), \( EH =? \). If \( \frac{EF}{AB} = \frac{7}{21} = \frac{1}{3} \), then \( EH = AD \times \frac{1}{3} = 6 \times \frac{1}{3} = 2 \)? No, the problem asks for \( \overline{GH} \). Wait, \( GH \) is opposite \( EF \), so \( GH = EF \)? No, in a parallelogram, opposite sides are equal, so \( EF = GH \), \( EH = FG \). Wait, that can't be. Wait, maybe the diagram is different. Wait, the first problem: Parallelogram \( ABCD \) with \( AB = 21 \) in, \( AD = 6 \) in; Parallelogram \( EFGH \) with \( EF = 7 \) in, \( EH =? \), and \( \overline{GH} \) is the side we need. Wait, maybe \( AB \) corresponds to \( GH \), and \( AD \) corresponds to \( EF \)? No, that doesn't make sense. Wait, let's re-express: similarity ratio is \( \frac{EF}{AD} = \frac{7}{6} \)? No, the problem must have \( AB = 21 \), \( EF = 7 \), so ratio \( 7/21 = 1/3 \). Then \( GH \) is equal to \( AB \times \) ratio? Wait, \( AB = 21 \), ratio \( 1/3 \), so \( GH = 21 \times (1/3) = 7 \)? Wait, but \( EF = 7 \), so \( GH = EF \)? No, in a parallelogram, \( EF = GH \), so that would be 7. Wait, maybe that's it. So \( \overline{GH} = 7 \) in? Wait, no, wait, \( AB = 21 \), \( EF = 7 \), so the ratio is \( 1/3 \), so \( GH = AB \times (1/3) = 7 \). So answer is 7?

Step 2: Wait, maybe I made a mistake. Let's check again. Parallelogram similarity: corresponding sides are proportional. So \( \frac{EF}{AB} = \frac{FG…

Step 1: Identify Corresponding Edges

The two rectangular prisms are similar. First prism: 8 in, 12 in, 3 in. Second prism: \( y \), 9 in, \( x \). So corresponding edges: 12 in (first) corresponds to 9 in (second)? Wait, 12 and 9: ratio \( 9/12 = 3/4 \). Then 8 in (first) corresponds to \( y \) (second): \( y = 8 \times (3/4) = 6 \) in. 3 in (first) corresponds to \( x \) (second): \( x = 3 \times (3/4) = 2.25 \)? Wait, no, wait: first prism dimensions: 8 (length), 12 (height), 3 (width). Second prism: \( y \) (length), 9 (height), \( x \) (width). So height ratio: \( 9/12 = 3/4 \). So length ratio: \( y/8 = 3/4 \) ⇒ \( y = 8 \times 3/4 = 6 \) in. Width ratio: \( x/3 = 3/4 \) ⇒ \( x = 3 \times 3/4 = 2.25 \)? Wait, no, maybe 12 is the height, 9 is the height, so ratio \( 9/12 = 3/4 \). Then length: 8 (first) → \( y \) (second): \( y = 8 \times 3/4 = 6 \). Width: 3 (first) → \( x \) (second): \( x = 3 \times 3/4 = 2.25 \)? Wait, but maybe the first prism is 8 (length), 12 (height), 3 (width); second is \( y \) (length), 9 (height), \( x \) (width). So height ratio: 9/12 = 3/4. So length: \( y = 8 \times 3/4 = 6 \), width: \( x = 3 \times 3/4 = 2.25 \)? Wait, but maybe I mixed up length and width. Alternatively, first prism: 8, 12, 3; second: \( y \), 9, \( x \). So 12 and 9: ratio 3/4. Then 8 and \( y \): \( y = 8 \times 3/4 = 6 \), 3 and \( x \): \( x = 3 \times 3/4 = 2.25 \). So \( x = 2.25 \) in, \( y = 6 \) in.

Step 2: Verify Proportions

Check ratios: \( 8/6 = 4/3 \), \( 12/9 = 4/3 \), \( 3/2.25 = 4/3 \). Yes, all ratios are \( 4/3 \) (or 3/4, depending on direction). So similarity holds.

Step 1: Identify Corresponding Edges

Triangular prisms: bottom prism has base 30 cm, height 8 cm, slant 20 cm? Wait, the bottom prism: base (let's say) 30 cm, height 8 cm, and the top prism: base \( x \), height 4 cm, slant \( y \). Similarity ratio: height ratio \( 4/8 = 1/2 \). So base ratio: \( x/30 = 1/2 \) ⇒ \( x = 15 \) cm. Slant ratio: \( y/20 = 1/2 \) ⇒ \( y = 10 \) cm.

Step 2: Verify Proportions

Height: \( 4/8 = 1/2 \), base: \( 15/30 = 1/2 \), slant: \( 10/20 = 1/2 \). All ratios 1/2, so similar.

Answer:

(Problem 1): \( \overline{GH} = 7 \) in? Wait, no, wait, maybe the ratio is \( AB/EF = 21/7 = 3 \), so \( GH = AB / 3 = 7 \). Yes.

Problem 2: Similar Rectangular Prisms (Find \( x \) and \( y \))