Question

1. multiple choice 1 point
2. rectangles abcd and wxyz are similar. which proportion must be true?

a (\frac{3}{5} = \frac{35}{x})
b (\frac{3}{x} = \frac{5}{35})
c (\frac{3}{x} = \frac{35}{5})
d (\frac{x}{3} = \frac{5}{35})
(there are two rectangles: rectangle abcd with length ab = 35 in, height ad = x in; rectangle wxyz with length wx = 5 in, height xy = 3 in)

Explanation:

Step1: Recall Similar Figures Property

For similar rectangles, the ratios of corresponding sides are equal. In rectangle \(ABCD\), sides are \(35\) in (length) and \(x\) in (width). In rectangle \(WXYZ\), sides are \(5\) in (length) and \(3\) in (width). So, \(\frac{\text{width of }WXYZ}{\text{width of }ABCD}=\frac{\text{length of }WXYZ}{\text{length of }ABCD}\) or \(\frac{3}{x}=\frac{5}{35}\)？Wait, no, let's match corresponding sides. \(AB = 35\) (length of \(ABCD\)), \(WX = 5\) (length of \(WXYZ\)); \(AD = x\) (width of \(ABCD\)), \(WZ = 3\) (width of \(WXYZ\)). So corresponding sides: \(AD\) corresponds to \(WZ\), \(AB\) corresponds to \(WX\). So \(\frac{AD}{WZ}=\frac{AB}{WX}\), which is \(\frac{x}{3}=\frac{35}{5}\)? No, wait, \(WZ\) is \(3\), \(AD\) is \(x\); \(AB\) is \(35\), \(WX\) is \(5\). Wait, maybe I mixed. Wait, rectangle \(WXYZ\) has length \(5\), width \(3\); rectangle \(ABCD\) has length \(35\), width \(x\). So the ratio of width to length for \(WXYZ\) is \(\frac{3}{5}\), and for \(ABCD\) is \(\frac{x}{35}\)? No, wait, similar figures: corresponding sides. So \(WX\) (length of small) corresponds to \(AB\) (length of large), \(WZ\) (width of small) corresponds to \(AD\) (width of large). So \(\frac{WZ}{AD}=\frac{WX}{AB}\), so \(\frac{3}{x}=\frac{5}{35}\)? Wait no, let's check options. Option B: \(\frac{3}{x}=\frac{5}{35}\)? Wait option B is \(\frac{3}{x}=\frac{5}{35}\)? Wait the options: A is \(\frac{3}{5}=\frac{35}{x}\), B is \(\frac{3}{x}=\frac{5}{35}\), C is \(\frac{3}{x}=\frac{35}{5}\), D is \(\frac{x}{3}=\frac{5}{35}\). Wait let's do correct proportion. Since the rectangles are similar, the ratio of corresponding sides is equal. So the width of the small rectangle (3) over the width of the large rectangle (x) should equal the length of the small rectangle (5) over the length of the large rectangle (35). So \(\frac{3}{x}=\frac{5}{35}\), which is option B? Wait no, wait option B is \(\frac{3}{x}=\frac{5}{35}\)? Wait the option B is written as \(\frac{3}{x}=\frac{5}{35}\)? Let me check the options again. The options: A: \(\frac{3}{5}=\frac{35}{x}\), B: \(\frac{3}{x}=\frac{5}{35}\), C: \(\frac{3}{x}=\frac{35}{5}\), D: \(\frac{x}{3}=\frac{5}{35}\). Wait, let's assign: Let’s denote rectangle 1: \(WXYZ\) with length \(WX = 5\), width \(XY = 3\). Rectangle 2: \(ABCD\) with length \(AB = 35\), width \(AD = x\). For similar rectangles, the ratio of length to width should be equal. So for \(WXYZ\), length/width = \(5/3\). For \(ABCD\), length/width = \(35/x\). So \(5/3 = 35/x\)? No, that's not an option. Wait, maybe width to length. For \(WXYZ\), width/length = \(3/5\). For \(ABCD\), width/length = \(x/35\). So \(3/5 = x/35\)? No, that's not. Wait, maybe corresponding sides: \(WX\) (5) corresponds to \(AB\) (35), \(XY\) (3) corresponds to \(AD\) (x). So the ratio of \(XY\) to \(AD\) is equal to the ratio of \(WX\) to \(AB\). So \(3/x = 5/35\), which is option B. Wait, let's check each option:

Option A: \(\frac{3}{5}=\frac{35}{x}\) → cross multiply: \(3x = 175\) → \(x = 175/3\). But let's see if that makes sense. If \(3/5 = 35/x\), then \(x = (35*5)/3 = 175/3 ≈ 58.33\). But let's see the other options.

Option B: \(\frac{3}{x}=\frac{5}{35}\) → cross multiply: \(5x = 105\) → \(x = 21\).

Option C: \(\frac{3}{x}=\frac{35}{5}\) → \(35x = 15\) → \(x = 15/35 = 3/7\), which is too small.

Option D: \(\frac{x}{3}=\frac{5}{35}\) → \(35x = 15\) → \(x = 15/35 = 3/7\), same as C.

Now, since the rectangles are similar, the ratio of corresponding sides should be equal. \(WX = 5\) (length of small), \(AB = 35\) (length of large); \(XY = 3\) (width of small), \(AD = x\) (width of large). So the ratio of width to length for small is \(3/5\), for large is \(x/35\). Wait, no, that would be \(3/5 = x/35\) → \(x = 21\), which is option B? Wait no, \(3/5 = x/35\) → \(x = (3*35)/5 = 21\). But option B is \(3/x = 5/35\), which is \(5x = 105\) → \(x = 21\). Oh, right, because \(3/x = 5/35\) is equivalent to \(3/5 = x/35\) (cross multiply: \(5x = 105\) → \(x = 21\); and \(3/5 = x/35\) → \(5x = 105\) → \(x = 21\)). So both are same. Wait, let's see the correspondence again. The small rectangle has length 5, width 3; large has length 35, width x. So the ratio of small width to large width is 3/x, and small length to large length is 5/35. Since they are similar, these ratios should be equal: 3/x = 5/35. So that's option B.

Step2: Verify the Proportion

For similar rectangles, corresponding sides are proportional. The side of length 3 in the small rectangle corresponds to the side of length \(x\) in the large rectangle, and the side of length 5 in the small rectangle corresponds to the side of length 35 in the large rectangle. Thus, \(\frac{3}{x}=\frac{5}{35}\) (ratio of corresponding widths = ratio of corresponding lengths).

Answer:

B. \(\frac{3}{x} = \frac{5}{35}\)