Question

8 multiple choice 1 point 8. trapezoids defg and jklm are similar. which proportion must be true? a $\frac{16}{x}=\frac{48}{12}$ b $\frac{48}{x}=\frac{16}{12}$ c $\frac{16}{x}=\frac{12}{48}$ d $\frac{12}{x}=\frac{16}{48}$

Explanation:

Step1: Recall similar figures property

For similar trapezoids, corresponding sides are proportional. So, side \( DG \) (48 m) in DEFG corresponds to side \( MJ \) (12 m) in JKLM, and side \( GF \) (x m) in DEFG corresponds to side \( ML \) (16 m) in JKLM.
Thus, the proportion is \( \frac{DG}{MJ}=\frac{GF}{ML} \), which is \( \frac{48}{12}=\frac{x}{16} \) or rearranged as \( \frac{48}{x}=\frac{16}{12} \) (by cross - multiplying and re - arranging the proportion \( \frac{48}{12}=\frac{x}{16} \) to get \( 48\times16 = 12x\), and then \( \frac{48}{x}=\frac{12}{16} \)? Wait, no, let's do it correctly.

Wait, let's identify corresponding sides. Trapezoid DEFG and JKLM are similar. So, \( DG = 48\), \( MJ=12\), \( GF = x\), \( ML = 16\). So the ratio of corresponding sides should be equal. So \( \frac{DG}{MJ}=\frac{GF}{ML} \) → \( \frac{48}{12}=\frac{x}{16} \). Cross - multiplying gives \( 48\times16=12x \). If we rearrange the proportion to solve for the ratio of \( \frac{48}{x}=\frac{16}{12} \) (by taking \( \frac{48}{x}=\frac{16}{12} \), cross - multiplying gives \( 48\times12 = 16x \), which is the same as \( 48\times16=12x \)? Wait, no, I made a mistake. Let's re - express.

From \( \frac{48}{12}=\frac{x}{16} \), we can cross - multiply: \( 48\times16 = 12x \). If we want to write a proportion with \( x \) in the denominator, let's solve for \( \frac{48}{x} \). From \( 48\times16=12x \), divide both sides by \( x\times12 \): \( \frac{48}{x}=\frac{12}{16} \)? No, that's not right. Wait, the options are:

A. \( \frac{16}{x}=\frac{48}{12} \)

B. \( \frac{48}{x}=\frac{16}{12} \)

C. \( \frac{16}{x}=\frac{12}{48} \)

D. \( \frac{12}{x}=\frac{16}{48} \)

Let's use the property of similar figures: corresponding sides are in proportion. So, side of DEFG (48) corresponds to side of JKLM (12), and side of DEFG (x) corresponds to side of JKLM (16). So the proportion is \( \frac{48}{12}=\frac{x}{16} \). Let's rearrange this proportion. Cross - multiply: \( 48\times16 = 12x \). Now, let's solve for \( \frac{48}{x} \). Divide both sides of \( 48\times16 = 12x \) by \( x\times12 \): \( \frac{48}{x}=\frac{12}{16} \)? No, that's not matching. Wait, maybe I mixed up the correspondence. Maybe \( DG = 48 \) corresponds to \( ML = 16 \), and \( GF=x \) corresponds to \( MJ = 12 \). Then the proportion is \( \frac{DG}{ML}=\frac{GF}{MJ} \), so \( \frac{48}{16}=\frac{x}{12} \). Cross - multiplying gives \( 48\times12=16x \), which can be rearranged as \( \frac{48}{x}=\frac{16}{12} \), which is option B.

Yes, that makes sense. If \( DG = 48 \) (DEFG) corresponds to \( ML = 16 \) (JKLM), and \( GF=x \) (DEFG) corresponds to \( MJ = 12 \) (JKLM), then \( \frac{DG}{ML}=\frac{GF}{MJ} \) → \( \frac{48}{16}=\frac{x}{12} \), and cross - multiplying gives \( 48\times12 = 16x \), which is equivalent to \( \frac{48}{x}=\frac{16}{12} \) (divide both sides of \( 48\times12 = 16x \) by \( x\times12 \)).

Step2: Analyze each option

- Option A: \( \frac{16}{x}=\frac{48}{12} \). Cross - multiplying gives \( 16\times12 = 48x\) → \( 192 = 48x\) → \( x = 4 \). But from the similar trapezoids, this ratio is incorrect.
- Option B: \( \frac{48}{x}=\frac{16}{12} \). Cross - multiplying gives \( 48\times12=16x\) → \( 576 = 16x\) → \( x = 36 \). Let's check with the correct proportion. If \( x = 36 \), then \( \frac{48}{36}=\frac{4}{3} \) and \( \frac{16}{12}=\frac{4}{3} \), so the ratios are equal.
- Option C: \( \frac{16}{x}=\frac{12}{48} \). Cross - multiplying gives \( 16\times48 = 12x\) → \( 768 = 12x\) → \( x = 64 \). The ratio \( \frac{16}{64}=\frac{1}{4} \) and \( \frac{12}{48}=\frac{1}{4} \), but the correspondence of sides is wrong.
- Option D: \( \frac{12}{x}=\frac{16}{48} \). Cross - multiplying gives \( 12\times48 = 16x\) → \( 576 = 16x\) → \( x = 36 \). But the ratio \( \frac{12}{36}=\frac{1}{3} \) and \( \frac{16}{48}=\frac{1}{3} \), but the correspondence of sides is wrong.

Only option B has the correct correspondence of sides for similar trapezoids.

Answer:

B. \( \frac{48}{x}=\frac{16}{12} \)