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A triangle has vertices at [tex]$A (-2,4)$[/tex], [tex]$B (-2,8)$[/tex], and [tex]$C (6,4)$[/tex]. If [tex]$A^{\prime}$[/tex] has coordinates of [tex]$(-0.25, 0.5)$[/tex] after the triangle has been dilated with a center of dilation about the origin, which statements are true? Select 3 options.

A. The coordinates of [tex]$C^{\prime}$[/tex] are [tex]$(0.75, 0.5)$[/tex].
B. The coordinates of [tex]$C^{\prime}$[/tex] are [tex]$(1.5, 1)$[/tex].
C. The scale factor is [tex]$\frac{1}{8}$[/tex].
D. The scale factor is 8.
E. The scale factor is [tex]$\frac{1}{4}$[/tex].
F. The scale factor is 4.
G. The coordinates of [tex]$B^{\prime}$[/tex] are [tex]$(-0.25, 1)$[/tex].
H. The coordinates of [tex]$B^{\prime}$[/tex] are [tex]$(-0.5, 2)$[/tex].

Sagot :

GinnyAnswer
Let's analyze the dilation process and verify which statements are true based on the given conditions and coordinates.

Step 1: Determine the scale factor

To find the scale factor of the dilation, we compare the coordinates of [tex]\( A \)[/tex] and [tex]\( A' \)[/tex]:

- Original coordinates of [tex]\( A \)[/tex] are [tex]\( (-2, 4) \)[/tex]
- Coordinates of [tex]\( A' \)[/tex] are [tex]\( (-0.25, 0.5) \)[/tex]

The scale factor [tex]\( k \)[/tex] can be calculated using the [tex]\( x \)[/tex]-coordinates:
[tex]\[ k = \frac{x_{A'}}{x_{A}} = \frac{-0.25}{-2} = 0.125 \][/tex]

Step 2: Verify the coordinates of [tex]\( B' \)[/tex] and [tex]\( C' \)[/tex] using the scale factor

- Original coordinates of [tex]\( B \)[/tex] are [tex]\( (-2, 8) \)[/tex]
- Original coordinates of [tex]\( C \)[/tex] are [tex]\( (6, 4) \)[/tex]

We apply the scale factor to find [tex]\( B' \)[/tex] and [tex]\( C' \)[/tex].

For [tex]\( B' \)[/tex]:
[tex]\[ B'_x = B_x \times k = -2 \times 0.125 = -0.25 \][/tex]
[tex]\[ B'_y = B_y \times k = 8 \times 0.125 = 1 \][/tex]

So, the coordinates of [tex]\( B' \)[/tex] are [tex]\( (-0.25, 1) \)[/tex].

For [tex]\( C' \)[/tex]:
[tex]\[ C'_x = C_x \times k = 6 \times 0.125 = 0.75 \][/tex]
[tex]\[ C'_y = C_y \times k = 4 \times 0.125 = 0.5 \][/tex]

So, the coordinates of [tex]\( C' \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex].

Step 3: Evaluate the given statements

- "The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (0.75, 0.5) \)[/tex]": True
- "The coordinates of [tex]\( C^{\prime} \)[/tex] are [tex]\( (1.5, 1) \)[/tex]": False
- "The scale factor is [tex]\( \frac{1}{8} \)[/tex]": True
- "The scale factor is 8": False
- "The scale factor is [tex]\( \frac{1}{4} \)[/tex]": False
- "The scale factor is 4": False
- "The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.25, 1) \)[/tex]": True
- "The coordinates of [tex]\( B^{\prime} \)[/tex] are [tex]\( (-0.5, 2) \)[/tex]": False

Conclusion:

Based on the above analysis, 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].
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