Question

2 rectangle dabc and rectangle dxyz 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, corresponding sides are proportional. Rectangle \(DABC\) has length \(30\) in and height \(20\) in. Rectangle \(DXYZ\) has length \(6\) in and height \(x\) in. So, \(\frac{\text{length of } DABC}{\text{length of } DXYZ}=\frac{\text{height of } DABC}{\text{height of } DXYZ}\), which is \(\frac{30}{6}=\frac{20}{x}\) (or equivalent proportion). Let's check each option:
- Option 1: \(\frac{30}{20}=\frac{6}{x}\) → Cross - multiply: \(30x = 120\) → \(x = 4\). But let's see proportion logic. Length of big rectangle is \(30\), height \(20\); length of small is \(6\), height \(x\). So \(\frac{30}{6}=\frac{20}{x}\) (length ratio = height ratio). Option 1 has \(\frac{30}{20}=\frac{6}{x}\) (length/height of big = length/height of small), which is incorrect.
- Option 2: \(\frac{30}{6}=\frac{x}{20}\) → Cross - multiply: \(30\times20=6x\) → \(600 = 6x\) → \(x = 100\). Incorrect proportion.
- Option 3: \(\frac{20}{30}=\frac{6}{x}\) → Cross - multiply: \(20x=180\) → \(x = 9\). Wait, let's re - express the correct proportion. The ratio of height of big to length of big should equal ratio of height of small to length of small? No, for similar rectangles, \(\frac{\text{length}_1}{\text{length}_2}=\frac{\text{height}_1}{\text{height}_2}\). So \(\frac{30}{6}=\frac{20}{x}\) can be rewritten as \(\frac{20}{30}=\frac{6}{x}\) (by cross - multiplying and rearranging: \(30\times6 = 20x\) → \(180 = 20x\) → \(x = 9\)? Wait, no, let's do it properly. If rectangle \(DABC\) (length \(30\), height \(20\)) and \(DXYZ\) (length \(6\), height \(x\)) are similar, then \(\frac{\text{length of } DABC}{\text{height of } DABC}=\frac{\text{length of } DXYZ}{\text{height of } DXYZ}\) is wrong. Correct is \(\frac{\text{length of } DABC}{\text{length of } DXYZ}=\frac{\text{height of } DABC}{\text{height of } DXYZ}\), so \(\frac{30}{6}=\frac{20}{x}\), which is equivalent to \(\frac{20}{30}=\frac{6}{x}\) (by taking reciprocals of both sides: \(\frac{6}{30}=\frac{x}{20}\) no, wait \(\frac{30}{6}=\frac{20}{x}\) → cross - multiply \(30x = 120\) → \(x = 4\)? Wait, I think I messed up the sides. Wait, in rectangle \(DABC\), the horizontal side (length) is \(AB = 30\) in, vertical side (height) is \(BC = 20\) in. In rectangle \(DXYZ\), horizontal side (length) is \(DZ = 6\) in, vertical side (height) is \(YZ=x\) in. So for similar rectangles, \(\frac{\text{length of } DABC}{\text{length of } DXYZ}=\frac{\text{height of } DABC}{\text{height of } DXYZ}\), so \(\frac{30}{6}=\frac{20}{x}\). Now let's check option 3: \(\frac{20}{30}=\frac{6}{x}\) → cross - multiply \(20x = 180\) → \(x = 9\). No, that's not right. Wait, maybe I mixed length and height. Wait, maybe the height of \(DABC\) is \(20\) (vertical), length is \(30\) (horizontal). The small rectangle \(DXYZ\): horizontal length is \(6\), vertical height is \(x\). So the ratio of horizontal to vertical in \(DABC\) is \(\frac{30}{20}\), and in \(DXYZ\) is \(\frac{6}{x}\). Since they are similar, \(\frac{30}{20}=\frac{6}{x}\) (option 1). Wait, now I'm confused. Let's use cross - multiplication for each option:
- Option 1: \(\frac{30}{20}=\frac{6}{x}\) → \(30x=20\times6 = 120\) → \(x = 4\)
- Option 3: \(\frac{20}{30}=\frac{6}{x}\) → \(20x = 30\times6=180\) → \(x = 9\)
- Let's think about the rectangles. The big rectangle has length \(30\) and width \(20\). The small rectangle has length \(6\) and width \(x\). Since they are similar, the ratio of length to width should be the same. So \(\frac{30}{20}=\frac{6}{x}\) (length/width of big = length/width of small). So option 1 is \(\frac{30}{20}=\frac{6}{x}\), which is correct. Wait, earlier mistake was in side identification. The vertical side of big rectangle is \(20\) (width), horizontal is \(30\) (length). Small rectangle: horizontal length \(6\), vertical width \(x\). So length/width for big: \(30/20\), length/width for small: \(6/x\). So they should be equal. So \(\frac{30}{20}=\frac{6}{x}\) is correct. Wait, but let's check option 3: \(\frac{20}{30}=\frac{6}{x}\) → width/length of big = length/width of small, which is not the same ratio. So the correct proportion is \(\frac{30}{20}=\frac{6}{x}\) (option 1) or let's re - express the proportion correctly.

Wait, maybe the labels: In rectangle \(DABC\), \(AB = 30\) (top side), \(BC = 20\) (right side). In rectangle \(DXYZ\), \(DZ = 6\) (bottom side), \(YZ=x\) (right side). So the sides \(AB\) (length of big) corresponds to \(DZ\) (length of small), and \(BC\) (height of big) corresponds to \(YZ\) (height of small). So \(\frac{AB}{DZ}=\frac{BC}{YZ}\) → \(\frac{30}{6}=\frac{20}{x}\), which is equivalent to \(\frac{20}{30}=\frac{6}{x}\) (by cross - multiplying and rearranging: \(30\times6=20x\) → \(180 = 20x\) → \(x = 9\))? No, \(\frac{30}{6}=\frac{20}{x}\) → \(30x=120\) → \(x = 4\). I think I made a mistake in side correspondence. Let's look at the figure: \(DABC\) is the big rectangle, \(DXYZ\) is the small one at the bottom left. So \(DA\) and \(BC\) are vertical sides (height \(20\) in), \(AB\) and \(DC\) are horizontal sides (length \(30\) in). \(DX\) and \(YZ\) are vertical sides (height \(x\) in), \(DZ\) and \(XY\) are horizontal sides (length \(6\) in). So for similar rectangles, the ratio of horizontal side to vertical side should be equal. So \(\frac{\text{horizontal of big}}{\text{vertical of big}}=\frac{\text{horizontal of small}}{\text{vertical of small}}\), so \(\frac{30}{20}=\frac{6}{x}\) (option 1). Let's check cross - multiplication: \(30x = 20\times6=120\) → \(x = 4\). If we use option 3: \(\frac{20}{30}=\frac{6}{x}\) → \(20x = 180\) → \(x = 9\). But let's see the figure again. The small rectangle's vertical side is \(x\), horizontal is \(6\). The big rectangle's vertical is \(20\), horizontal is \(30\). So the proportion of horizontal to vertical in big is \(30/20\), in small is \(6/x\). So they should be equal. So \(\frac{30}{20}=\frac{6}{x}\) is correct. Wait, but let's check the answer options again. Wait, maybe the correct proportion is \(\frac{20}{30}=\frac{6}{x}\). Wait, no, let's use the definition of similar polygons: corresponding sides are proportional. So side \(AB\) (length \(30\)) corresponds to side \(XY\) (length \(6\))? No, \(AB\) is from \(A\) to \(B\), \(XY\) is from \(X\) to \(Y\). Wait, the rectangles share the vertex \(D\). So \(DABC\): \(D\) to \(A\) to \(B\) to \(C\) to \(D\). \(DXYZ\): \(D\) to \(X\) to \(Y\) to \(Z\) to \(D\). So \(DA\) is vertical (length \(20\)), \(AB\) is horizontal (length \(30\)), \(BC\) is vertical (length \(20\)), \(CD\) is horizontal (length \(30\)). \(DX\) is vertical (length \(x\)), \(XY\) is horizontal (length \(6\)), \(YZ\) is vertical (length \(x\)), \(ZD\) is horizontal (length \(6\)). So the corresponding sides: \(DA\) (vertical, \(20\)) corresponds to \(DX\) (vertical, \(x\)), and \(AB\) (horizontal, \(30\)) corresponds to \(XY\) (horizontal, \(6\)). So the ratio of \(AB\) to \(DA\) should equal ratio of \(XY\) to \(DX\), so \(\frac{AB}{DA}=\frac{XY}{DX}\) → \(\frac{30}{20}=\frac{6}{x}\), which is option 1. Wait, but let's check the third option: \(\frac{20}{30}=\frac{6}{x}\) → \(\frac{DA}{AB}=\frac{XY}{DX}\), which is the reciprocal of the correct ratio. So the correct proportion is \(\frac{30}{20}=\frac{6}{x}\) (option 1) or \(\frac{20}{30}=\frac{6}{x}\) (option 3)? Wait, no, let's do cross - multiplication for option 3: \(\frac{20}{30}=\frac{6}{x}\) → \(20x = 180\) → \(x = 9\). For option 1: \(\frac{30}{20}=\frac{6}{x}\) → \(30x = 120\) → \(x = 4\). Which one makes sense? Let's think about the size. The small rectangle is inside the big one. The big rectangle is \(30\times20\), the small is \(6\times x\). If \(x = 4\), then the small rectangle is \(6\times4\), which is smaller than the big one's height of \(20\), which makes sense. If \(x = 9\), \(6\times9 = 54\), which is larger than the big rectangle's height of \(20\), which is impossible. So option 1 is wrong? Wait, I'm confused. Wait, maybe the corresponding sides are \(AB\) (length \(30\)) and \(DZ\) (length \(6\)), and \(BC\) (height \(20\)) and \(YZ\) (height \(x\)). So \(\frac{AB}{DZ}=\frac{BC}{YZ}\) → \(\frac{30}{6}=\frac{20}{x}\) → \(30x=120\) → \(x = 4\). But \(YZ\) is the vertical side of the small rectangle, so \(x = 4\) is less than \(20\), which is correct. Now let's check the options again. Option 3 is \(\frac{20}{30}=\frac{6}{x}\) → \(\frac{BC}{AB}=\frac{DZ}{YZ}\), which is the ratio of height to length of big rectangle equals ratio of length to height of small rectangle, which is not the same as corresponding sides. So the correct proportion is \(\frac{30}{6}=\frac{20}{x}\), which can be rewritten as \(\frac{20}{30}=\frac{6}{x}\) (by cross - multiplying: \(30\times6 = 20x\))? No, \(\frac{30}{6}=\frac{20}{x}\) cross - multiplies to \(30x = 120\), \(\frac{20}{30}=\frac{6}{x}\) cross - multiplies to \(20x = 180\). So there's a mistake in my side correspondence. Wait, the problem says "Rectangle \(DABC\) and rectangle \(DXYZ\) are similar". So the order of the letters matters. \(DABC\): \(D - A - B - C - D\), \(DXYZ\): \(D - X - Y - Z - D\). So the sides: \(DA\) corresponds to \(DX\), \(AB\) corresponds to \(XY\), \(BC\) corresponds to \(YZ\), \(CD\) corresponds to \(ZD\). So \(DA\) (vertical, length \(20\)) corresponds to \(DX\) (vertical, length \(x\)), \(AB\) (horizontal, length \(30\)) corresponds to \(XY\) (horizontal, length \(6\)). So the ratio of \(AB\) to \(DA\) is \(\frac{30}{20}\), and the ratio of \(XY\) to \(DX\) is \(\frac{6}{x}\). Since they are similar, \(\frac{AB}{DA}=\frac{XY}{DX}\) → \(\frac{30}{20}=\frac{6}{x}\), which is option 1. But when we calculate \(x = 4\), the small rectangle has dimensions \(6\times4\), which is inside the big rectangle (which is \(30\times20\)), so that makes sense. The other options: Option 3: \(\frac{20}{30}=\frac{6}{x}\) → \(x = 9\), which would make the small rectangle \(6\times9\), which is taller than the big rectangle's height of \(20\)? No, \(9\) is less than \(20\), wait \(6\times9 = 54\) in area, but the height is \(9\), which is less than \(20\). Wait, I think I messed up the height and length. Let's define length as horizontal and height as vertical. Big rectangle: length \(30\), height \(20\). Small rectangle: length \(6\), height \(x\). For similar rectangles, \(\frac{\text{length of big}}{\text{height of big}}=\frac{\text{length of small}}{\text{height of small}}\), so \(\frac{30}{20}=\frac{6}{x}\) (option 1) is correct.

Answer:

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