Question

1. figure f is a scaled copy of figure e.

we know:

• ab = 6
• cd = 3
• xy = 4
• zw = a

select all true equations.
a. \\( \frac { 6 } { 3 } = \frac { 4 } { a } \\)
b. \\( \frac { 6 } { 4 } = \frac { 3 } { a } \\)
c. \\( \frac { 3 } { 4 } = \frac { 6 } { a } \\)
d. \\( \frac { 6 } { 3 } = \frac { a } { 4 } \\)
e. \\( \frac { 6 } { 4 } = \frac { a } { 3 } \\)
f. \\( \frac { 3 } { 4 } = \frac { a } { 6 } \\)

Explanation:

Step1: Understand Scaled Copies

In scaled copies, corresponding lengths are proportional. For Figure \( e \) and Figure \( f \), \( AB \) corresponds to \( XY \), and \( CD \) corresponds to \( ZW \). So, the ratio of \( AB \) to \( CD \) should equal the ratio of \( XY \) to \( ZW \).

Step2: Set Up Proportion

Given \( AB = 6 \), \( CD = 3 \), \( XY = 4 \), \( ZW = a \). The proportion is \( \frac{AB}{CD}=\frac{XY}{ZW} \), which substitutes to \( \frac{6}{3}=\frac{4}{a} \) (Option A) and also can be rearranged. Let's check each option:

- Option A: \( \frac{6}{3}=\frac{4}{a} \) – This is the correct proportion (corresponding lengths).
- Option D: \( \frac{6}{3}=\frac{a}{4} \) – Cross - multiply: \( 6\times4 = 3\times a\Rightarrow24 = 3a\Rightarrow a = 8 \). From Option A: \( \frac{6}{3}=\frac{4}{a}\Rightarrow6a=12\Rightarrow a = 2 \). Contradicts, so D is wrong.
- Option B: \( \frac{6}{4}=\frac{3}{a} \) – Cross - multiply: \( 6a = 12\Rightarrow a = 2 \). Wait, let's re - check. Wait, \( AB = 6 \), \( XY = 4 \); \( CD = 3 \), \( ZW = a \). So \( \frac{AB}{XY}=\frac{CD}{ZW}\Rightarrow\frac{6}{4}=\frac{3}{a} \), which is Option B. Wait, maybe I made a mistake earlier. Let's re - establish the proportion. The ratio of the original figure (e) to the scaled figure (f) can be \( \frac{\text{Length in }e}{\text{Length in }f}=\frac{\text{Length in }e}{\text{Length in }f} \). So \( AB \) (e) is 6, \( XY \) (f) is 4; \( CD \) (e) is 3, \( ZW \) (f) is \( a \). So \( \frac{AB}{XY}=\frac{CD}{ZW}\Rightarrow\frac{6}{4}=\frac{3}{a} \) (Option B) and \( \frac{AB}{CD}=\frac{XY}{ZW}\Rightarrow\frac{6}{3}=\frac{4}{a} \) (Option A). Also, from \( \frac{6}{3}=\frac{4}{a} \), we can cross - multiply to get \( 6a = 12\Rightarrow a = 2 \). From \( \frac{6}{4}=\frac{3}{a} \), cross - multiply: \( 6a = 12\Rightarrow a = 2 \). Now check Option E: \( \frac{6}{4}=\frac{a}{3} \) – Cross - multiply: \( 18 = 4a\Rightarrow a=\frac{9}{2} \), wrong. Option C: \( \frac{3}{4}=\frac{6}{a} \) – Cross - multiply: \( 3a = 24\Rightarrow a = 8 \), wrong. Option F: \( \frac{3}{4}=\frac{a}{6} \) – Cross - multiply: \( 18 = 4a\Rightarrow a=\frac{9}{2} \), wrong. Wait, let's re - do the proportion. Since Figure \( f \) is a scaled copy of Figure \( e \), the scale factor from \( e \) to \( f \) is \( \frac{XY}{AB}=\frac{4}{6}=\frac{2}{3} \). So \( ZW = CD\times\frac{2}{3}=3\times\frac{2}{3}=2 \). Now check Option A: \( \frac{6}{3}=\frac{4}{a}\Rightarrow\frac{6}{3}=\frac{4}{2}\Rightarrow2 = 2 \), correct. Option B: \( \frac{6}{4}=\frac{3}{a}\Rightarrow\frac{6}{4}=\frac{3}{2}\Rightarrow\frac{3}{2}=\frac{3}{2} \), correct. Wait, the original problem says "Select all true equations". Let's re - analyze:

The two corresponding pairs are \( (AB, XY) \) and \( (CD, ZW) \). So the proportion can be written as \( \frac{AB}{CD}=\frac{XY}{ZW} \) (A) or \( \frac{AB}{XY}=\frac{CD}{ZW} \) (B) or \( \frac{CD}{AB}=\frac{ZW}{XY} \) (not in options) or \( \frac{CD}{ZW}=\frac{AB}{XY} \) (B is \( \frac{AB}{XY}=\frac{CD}{ZW} \)). Let's check the values:

For A: \( \frac{6}{3}=\frac{4}{a}\Rightarrow a = 2 \). For B: \( \frac{6}{4}=\frac{3}{a}\Rightarrow6a = 12\Rightarrow a = 2 \). So both A and B are correct? Wait, the options given:

Wait the user's figure: Options are A. \( \frac{6}{3}=\frac{4}{a} \), B. \( \frac{6}{4}=\frac{3}{a} \), C. \( \frac{3}{4}=\frac{6}{a} \), D. \( \frac{6}{3}=\frac{a}{4} \), E. \( \frac{6}{4}=\frac{a}{3} \), F. \( \frac{3}{4}=\frac{a}{6} \).

Wait let's recalculate:

From scaled copy, \( \frac{AB}{XY}=\frac{CD}{ZW} \) (since \( AB \) and \( XY \) are corresponding, \( CD \) and \( ZW \) are corresponding). So \( \frac{6}{4}=\frac{3}{a} \) (Option B) and \( \frac{AB}{CD}=\frac{XY}{ZW}\Rightarrow\frac{6}{3}=\frac{4}{a} \) (Option A). Let's check with \( a = 2 \):

Option A: \( \frac{6}{3}=2 \), \( \frac{4}{2}=2 \), so A is true.

Option B: \( \frac{6}{4}=\frac{3}{2}=1.5 \), so B is true.

Wait but maybe I made a mistake in the proportion direction. Wait, Figure \( f \) is a scaled copy of Figure \( e \), so Figure \( e \) is the original, Figure \( f \) is the copy. So the scale factor is \( \frac{\text{length in }f}{\text{length in }e} \). So \( XY=\text{scale factor}\times AB \), \( ZW=\text{scale factor}\times CD \). So \( \text{scale factor}=\frac{XY}{AB}=\frac{4}{6}=\frac{2}{3} \). Then \( ZW=\frac{2}{3}\times CD=\frac{2}{3}\times3 = 2 \). So \( a = 2 \).

Now check each option:

A. \( \frac{6}{3}=\frac{4}{a}\Rightarrow2=\frac{4}{2}\Rightarrow2 = 2 \): True.

B. \( \frac{6}{4}=\frac{3}{a}\Rightarrow\frac{3}{2}=\frac{3}{2}\): True.

Wait but the original problem's options – maybe the user made a typo, but according to the calculation, A and B are true. Wait, but let's check the options again. Wait the user's options:

A. \( \frac{6}{3}=\frac{4}{a} \)

B. \( \frac{6}{4}=\frac{3}{a} \)

C. \( \frac{3}{4}=\frac{6}{a} \)

D. \( \frac{6}{3}=\frac{a}{4} \)

E. \( \frac{6}{4}=\frac{a}{3} \)

F. \( \frac{3}{4}=\frac{a}{6} \)

Wait, maybe the intended correct options are A and B? But let's re - check the proportion. The key is that in similar figures (scaled copies), corresponding sides are proportional. So \( AB \) (6) and \( CD \) (3) are in Figure \( e \); \( XY \) (4) and \( ZW \) (a) are in Figure \( f \). So the ratio of \( AB \) to \( CD \) is equal to the ratio of \( XY \) to \( ZW \), so \( \frac{AB}{CD}=\frac{XY}{ZW}\Rightarrow\frac{6}{3}=\frac{4}{a} \) (A is true). Also, \( \frac{AB}{XY}=\frac{CD}{ZW}\Rightarrow\frac{6}{4}=\frac{3}{a} \) (B is true). Let's check D: \( \frac{6}{3}=\frac{a}{4}\Rightarrow a = 8 \), which is wrong. E: \( \frac{6}{4}=\frac{a}{3}\Rightarrow a=\frac{18}{4}=4.5 \), wrong. C: \( \frac{3}{4}=\frac{6}{a}\Rightarrow a = 8 \), wrong. F: \( \frac{3}{4}=\frac{a}{6}\Rightarrow a = 4.5 \), wrong. So the true equations are A and B. But maybe the problem has a different intended answer. Wait, maybe I mixed up the correspondence. Wait, \( CD \) is 3 (vertical in Figure \( e \)), \( ZW \) is \( a \) (vertical in Figure \( f \)); \( AB \) is 6 (horizontal in \( e \)), \( XY \) is 4 (horizontal in \( f \)). So horizontal lengths: \( AB = 6 \), \( XY = 4 \); vertical lengths: \( CD = 3 \), \( ZW = a \). So the ratio of horizontal to vertical in \( e \) should equal the ratio in \( f \). So \( \frac{AB}{CD}=\frac{XY}{ZW}\Rightarrow\frac{6}{3}=\frac{4}{a} \) (A) and \( \frac{CD}{AB}=\frac{ZW}{XY}\Rightarrow\frac{3}{6}=\frac{a}{4}\Rightarrow\frac{1}{2}=\frac{a}{4}\Rightarrow a = 2 \), which is consistent with A. Also, \( \frac{AB}{XY}=\frac{CD}{ZW}\Rightarrow\frac{6}{4}=\frac{3}{a}\Rightarrow a = 2 \), consistent with B. So both A and B are true. But let's check the options again. If the problem is from a textbook, maybe the intended answers are A and B. But let's confirm with the proportion.

Answer:

A. \(\frac{6}{3}=\frac{4}{a}\), B. \(\frac{6}{4}=\frac{3}{a}\)