Answered

Discover new perspectives and gain insights with IDNLearn.com. Our platform provides prompt, accurate answers from experts ready to assist you with any question you may have.

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 :

GinnyAnsweridn
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].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.