Question

2 rectangle dabc and rectangle oxyz are similar.
which proportion must be true?
\\(\frac{30}{20} = \frac{6}{x}\\)
\\(\frac{30}{6} = \frac{x}{20}\\)
\\(\frac{20}{30} = \frac{6}{x}\\)
\\(\frac{6}{30} = \frac{20}{x}\\)

Explanation:

Step1: Recall Similar Figures Property

For similar rectangles, the ratios of corresponding sides are equal. Rectangle \( DABC \) has length \( 30 \) in and height \( 20 \) in. Rectangle \( OXYZ \) (the gray one) has height \( 6 \) in and length related to \( x \). The corresponding sides should be in proportion: \(\frac{\text{Length of } DABC}{\text{Height of } DABC}=\frac{\text{Length of } OXYZ}{\text{Height of } OXYZ}\) or \(\frac{\text{Height of } DABC}{\text{Length of } DABC}=\frac{\text{Height of } OXYZ}{\text{Length of } OXYZ}\). Wait, actually, for similar rectangles, \(\frac{\text{Length of larger rectangle}}{\text{Height of larger rectangle}}=\frac{\text{Length of smaller rectangle}}{\text{Height of smaller rectangle}}\). Wait, the larger rectangle \( DABC \) has length \( 30 \) and height \( 20 \). The smaller rectangle \( OXYZ \) has height \( 6 \) and length (let's see, the base of smaller is \( DZ \), but wait, the horizontal side: wait, maybe the length of larger is \( 30 \), height \( 20 \); smaller has height \( 6 \), and the length of smaller's base (horizontal) is such that the ratio of length to height is same. Wait, the options: let's check each.

Option 1: \(\frac{30}{20}=\frac{6}{x}\) → cross multiply: \(30x = 120\) → \(x = 4\). But let's think about similar rectangles. The larger rectangle has length \( 30 \), height \( 20 \). The smaller has height \( 6 \), and the length of the smaller's base (horizontal) is... Wait, maybe the ratio of length to height for larger is \( 30/20 \), and for smaller, the length (let's say the horizontal side of smaller is \( z \), but in the diagram, the gray rectangle has height \( 6 \) and the horizontal side? Wait, no, the gray rectangle is \( OXYZ \), with height \( 6 \) (vertical) and length (horizontal) such that the remaining horizontal is \( x \). Wait, the total length of \( DABC \) is \( 30 \), so the length of \( OXYZ \) plus \( x \) is \( 30 \)? No, wait, the diagram: \( DABC \) is a rectangle with \( DA \) and \( BC \) as vertical sides (height \( 20 \)), \( AB \) and \( DC \) as horizontal (length \( 30 \)). The gray rectangle \( OXYZ \) has \( DX \) and \( YZ \) as vertical (height \( 6 \)), \( DZ \) and \( XY \) as horizontal. So the larger rectangle has length \( 30 \), height \( 20 \); smaller has length \( DZ \) (let's say \( l \)) and height \( 6 \). Since they are similar, \(\frac{\text{Length of larger}}{\text{Height of larger}}=\frac{\text{Length of smaller}}{\text{Height of smaller}}\) → \(\frac{30}{20}=\frac{l}{6}\)? No, wait, maybe the other way: \(\frac{\text{Height of larger}}{\text{Length of larger}}=\frac{\text{Height of smaller}}{\text{Length of smaller}}\). Wait, no, similar figures have corresponding sides proportional. So the vertical side of larger is \( 20 \), horizontal is \( 30 \); vertical side of smaller is \( 6 \), horizontal is (let's say) \( s \). So \(\frac{20}{30}=\frac{6}{s}\) → which is \(\frac{20}{30}=\frac{6}{x}\)? Wait, no, the options: let's check each.

Wait, the correct proportion should be \(\frac{\text{Length of larger}}{\text{Height of larger}}=\frac{\text{Length of smaller}}{\text{Height of smaller}}\). Wait, larger length is \( 30 \), larger height \( 20 \); smaller height is \( 6 \), smaller length (the horizontal side of smaller) is... Wait, maybe the smaller rectangle's length is \( 6 \)? No, the smaller has height \( 6 \), and the horizontal side is such that the ratio of length to height is same as larger. Wait, larger ratio: length/height = 30/20 = 3/2. So smaller should have length/height = 3/2. So if smaller height is 6, length is (3/2)*6 = 9? No, that doesn't fit. Wait, maybe the other way: height/length = 20/30 = 2/3. So smaller height/length = 2/3 → 6/s = 2/3 → s = 9. Then x would be 30 - 9 = 21? No, that's not matching. Wait, maybe the proportion is \(\frac{20}{30}=\frac{6}{x}\) → which is option 3: \(\frac{20}{30}=\frac{6}{x}\). Let's check: cross multiply 20x = 180 → x = 9. Then the length of smaller is 9, height 6. Then 9/6 = 3/2, and 30/20 = 3/2. Yes! So that's the correct proportion. So \(\frac{20}{30}=\frac{6}{x}\) is correct. Let's verify:

Larger rectangle: length 30, height 20 → ratio height/length = 20/30 = 2/3.

Smaller rectangle: height 6, length (let's say the horizontal side of smaller is 9, then x is 30 - 9 = 21? Wait, no, the diagram: \( DZ \) is the length of the smaller rectangle, and \( ZC \) is \( x \). So total length \( DZ + x = 30 \). So if \( DZ \) is 9, then \( x = 21 \). But the proportion \(\frac{20}{30}=\frac{6}{x}\) → x = (6*30)/20 = 9. Wait, no, cross multiply: 20x = 30*6 → 20x = 180 → x = 9. Oh, right! So the length of the smaller rectangle's base (DZ) is 9? No, wait, the proportion is \(\frac{\text{Height of larger}}{\text{Length of larger}}=\frac{\text{Height of smaller}}{\text{Length of smaller}}\). So height of larger is 20, length 30; height of smaller is 6, length of smaller is x? Wait, no, maybe I got the sides reversed. Let's think again.

Similar rectangles: corresponding sides are proportional. So the length of the larger rectangle (30) corresponds to the length of the smaller rectangle (let's say the horizontal side of the smaller is, but in the diagram, the gray rectangle has height 6 (vertical) and the horizontal side is such that the ratio of length to height is same as the larger. The larger has length 30 (horizontal) and height 20 (vertical). So length/height = 30/20. The smaller has length (let's say the horizontal side of the smaller is y) and height 6 (vertical). So 30/20 = y/6 → y = (30*6)/20 = 9. Then the remaining horizontal length is x = 30 - 9 = 21? No, but the options: let's check each option.

Option 1: 30/20 = 6/x → cross multiply: 30x = 120 → x=4. Not 9.

Option 2: 30/6 = x/20 → cross multiply: 6x=600 → x=100. No.

Option 3: 20/30 = 6/x → cross multiply: 20x=180 → x=9. Which matches our previous calculation.

Option 4: 6/30 = 20/x → cross multiply: 6x=600 → x=100. No.

So the correct proportion is \(\frac{20}{30}=\frac{6}{x}\), which is the third option (the one with 20/30 = 6/x).

Answer:

\(\boldsymbol{\frac{20}{30}=\frac{6}{x}}\) (the option with \(\frac{20}{30}=\frac{6}{x}\))