Question

rectangle efgh is similar to rectangle jklm. which proportion can be used to find the value of x? (image of two rectangles: efgh with length 5 and width 2; jklm with length 12 and width x) options: 2/12 = 5/x, 5/2 = x/12, 5/12 = 2/x, 12/5 = 2/x

Explanation:

Step1: Recall Similar Figures Property

For similar rectangles, the ratios of corresponding sides are equal. In rectangle \(EFGH\), the sides are \(5\) (length) and \(2\) (width). In rectangle \(JKLM\), the length is \(12\) and the width is \(x\). So the ratio of length to width in \(EFGH\) should equal the ratio of length to width in \(JKLM\).

Step2: Set Up the Proportion

The length of \(EFGH\) is \(5\), width is \(2\); length of \(JKLM\) is \(12\), width is \(x\). So \(\frac{\text{Length of } EFGH}{\text{Width of } EFGH}=\frac{\text{Length of } JKLM}{\text{Width of } JKLM}\), which gives \(\frac{5}{2}=\frac{12}{x}\)? Wait, no, wait. Wait, maybe I mixed up. Wait, \(EFGH\) has length \(5\), width \(2\); \(JKLM\) has length \(12\), width \(x\). Wait, actually, the corresponding sides: \(EF = 5\) corresponds to \(JK = 12\), and \(FG = 2\) corresponds to \(KL = x\). So the ratio of \(EF\) to \(FG\) (in \(EFGH\)) should equal the ratio of \(JK\) to \(KL\) (in \(JKLM\)). So \(\frac{EF}{FG}=\frac{JK}{KL}\), so \(\frac{5}{2}=\frac{12}{x}\)? No, the options: let's check the options. The options are \(\frac{2}{12}=\frac{5}{x}\), \(\frac{5}{2}=\frac{x}{12}\), \(\frac{5}{12}=\frac{2}{x}\), \(\frac{12}{5}=\frac{2}{x}\). Wait, maybe I got the correspondence wrong. Let's see: \(EFGH\) has sides \(5\) (horizontal) and \(2\) (vertical). \(JKLM\) has horizontal side \(12\) and vertical side \(x\). So the ratio of horizontal to vertical in \(EFGH\) is \(\frac{5}{2}\), and in \(JKLM\) is \(\frac{12}{x}\)? No, that can't be. Wait, maybe the other way: horizontal of \(EFGH\) to horizontal of \(JKLM\) and vertical of \(EFGH\) to vertical of \(JKLM\). So \(\frac{5}{12}=\frac{2}{x}\). Wait, let's check: if \(EFGH\) and \(JKLM\) are similar, then \(\frac{\text{length of } EFGH}{\text{length of } JKLM}=\frac{\text{width of } EFGH}{\text{width of } JKLM}\). So \(\frac{5}{12}=\frac{2}{x}\), which is the third option? Wait, no, the third option is \(\frac{5}{12}=\frac{2}{x}\). Wait, let's re-express. Let's list the sides:

Rectangle \(EFGH\): length \(l_1 = 5\), width \(w_1 = 2\)

Rectangle \(JKLM\): length \(l_2 = 12\), width \(w_2 = x\)

For similar rectangles, \(\frac{l_1}{w_1}=\frac{l_2}{w_2}\) or \(\frac{l_1}{l_2}=\frac{w_1}{w_2}\)

First, \(\frac{l_1}{l_2}=\frac{w_1}{w_2}\) gives \(\frac{5}{12}=\frac{2}{x}\), which is the third option.

Wait, let's check the options:

1. \(\frac{2}{12}=\frac{5}{x}\) → cross multiply: \(2x = 60\) → \(x = 30\)

2. \(\frac{5}{2}=\frac{x}{12}\) → cross multiply: \(2x = 60\) → \(x = 30\)? No, wait, \(\frac{5}{2}=\frac{x}{12}\) → \(x=\frac{5\times12}{2}=30\)

3. \(\frac{5}{12}=\frac{2}{x}\) → cross multiply: \(5x = 24\) → \(x=\frac{24}{5}=4.8\)

4. \(\frac{12}{5}=\frac{2}{x}\) → cross multiply: \(12x = 10\) → \(x=\frac{10}{12}=\frac{5}{6}\)

Wait, but let's think again. The correct correspondence: \(EF = 5\) (top side of \(EFGH\)) corresponds to \(JK = 12\) (top side of \(JKLM\)), and \(FG = 2\) (right side of \(EFGH\)) corresponds to \(KL = x\) (right side of \(JKLM\)). So the ratio of \(EF\) to \(FG\) (in \(EFGH\)) should equal the ratio of \(JK\) to \(KL\) (in \(JKLM\)). So \(\frac{EF}{FG}=\frac{JK}{KL}\) → \(\frac{5}{2}=\frac{12}{x}\)? But that's not an option. Wait, maybe the ratio of \(EF\) to \(JK\) equals \(FG\) to \(KL\). So \(\frac{EF}{JK}=\frac{FG}{KL}\) → \(\frac{5}{12}=\frac{2}{x}\), which is the third option (\(\frac{5}{12}=\frac{2}{x}\)). Let's verify:

If \(\frac{5}{12}=\frac{2}{x}\), then \(5x = 24\), so \(x = \frac{24}{5}=4.8\). Let's check the other way: if we take \(\frac{5}{2}=\frac{12}{x}\), then \(5x = 24\), same \(x\). Wait, no, \(\frac{5}{2}=\frac{12}{x}\) is equivalent to \(\frac{5}{12}=\frac{2}{x}\) (cross - multiply: \(5x = 24\) and \(5x = 24\)). Wait, the option \(\frac{5}{12}=\frac{2}{x}\) is one of the options (the third one). Wait, but let's check the options again. The options are:

1. \(\frac{2}{12}=\frac{5}{x}\)

2. \(\frac{5}{2}=\frac{x}{12}\)

3. \(\frac{5}{12}=\frac{2}{x}\)

4. \(\frac{12}{5}=\frac{2}{x}\)

Wait, let's re - express the proportion correctly. For similar rectangles, corresponding sides are proportional. So the length of the first rectangle over the length of the second rectangle equals the width of the first rectangle over the width of the second rectangle.

First rectangle (\(EFGH\)): length \(= 5\), width \(= 2\)

Second rectangle (\(JKLM\)): length \(= 12\), width \(= x\)

So \(\frac{\text{Length of } EFGH}{\text{Length of } JKLM}=\frac{\text{Width of } EFGH}{\text{Width of } JKLM}\)

\(\frac{5}{12}=\frac{2}{x}\)

Yes, that's the third option.

Wait, but let's check the second option: \(\frac{5}{2}=\frac{x}{12}\). Cross - multiply: \(5\times12 = 2x\) → \(60 = 2x\) → \(x = 30\), which is not correct.

First option: \(\frac{2}{12}=\frac{5}{x}\) → \(2x = 60\) → \(x = 30\), wrong.

Fourth option: \(\frac{12}{5}=\frac{2}{x}\) → \(12x = 10\) → \(x=\frac{5}{6}\), wrong.

Third option: \(\frac{5}{12}=\frac{2}{x}\) → \(5x = 24\) → \(x = 4.8\), which makes sense. So the correct proportion is \(\frac{5}{12}=\frac{2}{x}\), which is the third option (the one with \(\frac{5}{12}=\frac{2}{x}\)). Wait, but let's check the labels again. Maybe \(EF = 5\) (length) and \(FG = 2\) (width) in \(EFGH\), and \(JK = 12\) (length) and \(KL = x\) (width) in \(JKLM\). So the ratio of length to width in \(EFGH\) is \(\frac{5}{2}\), and in \(JKLM\) is \(\frac{12}{x}\). So \(\frac{5}{2}=\frac{12}{x}\), which is equivalent to \(\frac{5}{12}=\frac{2}{x}\) (by cross - multiplying and rearranging). So the correct proportion is \(\frac{5}{12}=\frac{2}{x}\), which is the third option.

Answer:

\(\frac{5}{12}=\frac{2}{x}\) (the third option, the one with \(\frac{5}{12}=\frac{2}{x}\))