Answer :
To solve this problem, we need to determine the scale factor of the dilation and then use this scale factor to find the positions of the other vertices after dilation.
1. Determine the scale factor:
The original coordinates of [tex]\( A \)[/tex] are [tex]\( (-2, 4) \)[/tex], and the coordinates of [tex]\( A^{\prime} \)[/tex] after dilation are [tex]\( (-0.25, 0.5) \)[/tex].
Calculate the scale factor for each coordinate component:
- For the x-coordinate:
[tex]\[ \text{scale factor}_x = \frac{-0.25}{-2} = 0.125 \][/tex]
- For the y-coordinate:
[tex]\[ \text{scale factor}_y = \frac{0.5}{4} = 0.125 \][/tex]
Since both scale factors are equal, the scale factor for the dilation is [tex]\( 0.125 \)[/tex].
2. Calculate the coordinates of [tex]\( C^{\prime} \)[/tex]:
The original coordinates of [tex]\( C \)[/tex] are [tex]\( (6, 4) \)[/tex].
- New x-coordinate:
[tex]\[ 6 \times 0.125 = 0.75 \][/tex]
- New y-coordinate:
[tex]\[ 4 \times 0.125 = 0.5 \][/tex]
Therefore, the coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex].
3. Calculate the coordinates of [tex]\( B^{\prime} \)[/tex]:
The original coordinates of [tex]\( B \)[/tex] are [tex]\( (-2, 8) \)[/tex].
- New x-coordinate:
[tex]\[ -2 \times 0.125 = -0.25 \][/tex]
- New y-coordinate:
[tex]\[ 8 \times 0.125 = 1.0 \][/tex]
Therefore, the coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex].
4. Analyze the given statements:
- The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex]:
[tex]\[ \text{True} \][/tex]
- The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (1.5, 1) \)[/tex]:
[tex]\[ \text{False} \][/tex]
- The scale factor is [tex]\( \frac{1}{8} \)[/tex]:
[tex]\[ \text{True} \quad \left( 0.125 = \frac{1}{8} \right) \][/tex]
- The scale factor is 8:
[tex]\[ \text{False} \][/tex]
- The scale factor is [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[ \text{False} \][/tex]
- The scale factor is 4:
[tex]\[ \text{False} \][/tex]
- The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex]:
[tex]\[ \text{True} \][/tex]
- The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.5, 2) \)[/tex]:
[tex]\[ \text{False} \][/tex]
Therefore, the three true statements are:
1. The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex].
2. The scale factor is [tex]\( \frac{1}{8} \)[/tex].
3. The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex].
1. Determine the scale factor:
The original coordinates of [tex]\( A \)[/tex] are [tex]\( (-2, 4) \)[/tex], and the coordinates of [tex]\( A^{\prime} \)[/tex] after dilation are [tex]\( (-0.25, 0.5) \)[/tex].
Calculate the scale factor for each coordinate component:
- For the x-coordinate:
[tex]\[ \text{scale factor}_x = \frac{-0.25}{-2} = 0.125 \][/tex]
- For the y-coordinate:
[tex]\[ \text{scale factor}_y = \frac{0.5}{4} = 0.125 \][/tex]
Since both scale factors are equal, the scale factor for the dilation is [tex]\( 0.125 \)[/tex].
2. Calculate the coordinates of [tex]\( C^{\prime} \)[/tex]:
The original coordinates of [tex]\( C \)[/tex] are [tex]\( (6, 4) \)[/tex].
- New x-coordinate:
[tex]\[ 6 \times 0.125 = 0.75 \][/tex]
- New y-coordinate:
[tex]\[ 4 \times 0.125 = 0.5 \][/tex]
Therefore, the coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex].
3. Calculate the coordinates of [tex]\( B^{\prime} \)[/tex]:
The original coordinates of [tex]\( B \)[/tex] are [tex]\( (-2, 8) \)[/tex].
- New x-coordinate:
[tex]\[ -2 \times 0.125 = -0.25 \][/tex]
- New y-coordinate:
[tex]\[ 8 \times 0.125 = 1.0 \][/tex]
Therefore, the coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex].
4. Analyze the given statements:
- The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex]:
[tex]\[ \text{True} \][/tex]
- The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (1.5, 1) \)[/tex]:
[tex]\[ \text{False} \][/tex]
- The scale factor is [tex]\( \frac{1}{8} \)[/tex]:
[tex]\[ \text{True} \quad \left( 0.125 = \frac{1}{8} \right) \][/tex]
- The scale factor is 8:
[tex]\[ \text{False} \][/tex]
- The scale factor is [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[ \text{False} \][/tex]
- The scale factor is 4:
[tex]\[ \text{False} \][/tex]
- The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex]:
[tex]\[ \text{True} \][/tex]
- The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.5, 2) \)[/tex]:
[tex]\[ \text{False} \][/tex]
Therefore, the three true statements are:
1. The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex].
2. The scale factor is [tex]\( \frac{1}{8} \)[/tex].
3. The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex].
