Question

Question
Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 2 to 8 and its y-axis ranges from negative 1 to 7, both by 1-unit increments. Point upper A, point upper B, and an unlabeled third point are plotted and connected by lines.

A dilation of AB¯¯¯¯¯¯¯¯
occurs using a scale factor of 12
and a center of dilation at the origin. Prove that A′B′¯¯¯¯¯¯¯¯¯¯
is half the length of AB¯¯¯¯¯¯¯¯
by finding the points A′
and B′
.

(1 point)
Responses

A′
is at (1,5)
and B′
is at (3,6)
.
upper A prime is at left parenthesis 1 comma 5 right parenthesis and upper B prime is at left parenthesis 3 comma 6 right parenthesis .

A′
is at (4,8)
and B′
is at (12,12)
.
upper A prime is at left parenthesis 4 comma 8 right parenthesis and upper B prime is at left parenthesis 12 comma 12 right parenthesis .

A′
is at (−1,−2)
and B′
is at (−3,−3)
.
upper A prime is at left parenthesis negative 1 comma negative 2 right parenthesis and upper B prime is at left parenthesis negative 3 comma negative 3 right parenthesis .

A′
is at (1,2)
and B′
is at (3,3)
.
upper A prime is at left parenthesis 1 comma 2 right parenthesis and upper B prime is at left parenthesis 3 comma 3 right parenthesis .
Skip to navigation






Answers

Answer
a=(4,2)B=(6,6)
Answer
whats the answer
Answered by GPT-4o mini
To find the coordinates of points A' and B' after the dilation using a scale factor of \( \frac{1}{2} \) (since the problem states that A'B' is half the length of AB), we apply the formula for dilation centered at the origin:

\[
(x', y') = (kx, ky)
\]

where \( k \) is the scale factor, and \( (x, y) \) are the original coordinates of the points.

Assuming point A is at \( (x_A, y_A) \) and point B is at \( (x_B, y_B) \), we calculate:

- The new coordinates for point A' will be \( A' = \left(\frac{1}{2}x_A, \frac{1}{2}y_A\right) \)
- The new coordinates for point B' will be \( B' = \left(\frac{1}{2}x_B, \frac{1}{2}y_B\right) \)

Since we are not provided with the actual coordinates of points A and B in this context, we cannot directly solve for A' and B'.

However, if we had, for example, that point A is at (2, 6) and point B is at (4, 8), the calculations would be:

- For A: \( A' = \left(\frac{1}{2} \times 2, \frac{1}{2} \times 6\right) = (1, 3) \)
- For B: \( B' = \left(\frac{1}{2} \times 4, \frac{1}{2} \times 8\right) = (2, 4) \)

Now, based on the potential responses provided:

1. A′ is at (1,5) and B′ is at (3,6).
2. A′ is at (4,8) and B′ is at (12,12).
3. A′ is at (−1,−2) and B′ is at (−3,−3).
4. A′ is at (1,2) and B′ is at (3,3).

Among these possible responses, we look for the scaling down by half.

**Conclusion:**
Without the concrete points A and B given in the image or the problem, if we apply the \( k = \frac{1}{2} \) correctly with random example points we might reach A' and B.' The correct response would depend on the original points. If the problem context provides \( A (2, 6) \) and \( B (4, 8) \) as an example, we might derive that the answer is A' = (1,3) and B' would not match best with the given response options.

Therefore, it's essential to note where the exact original points are to properly assess A' and B', but in general based on example:

**Please replace the coordinates for A and B according to the image to derive the suitable A' and B'.**
Answered by GPT-4o mini
Given the points \( A = (4, 2) \) and \( B = (6, 6) \), we can find the coordinates of \( A' \) and \( B' \) after the dilation with a scale factor of \( \frac{1}{2} \) using the formula:

\[
(x', y') = \left( \frac{1}{2} x, \frac{1}{2} y \right)
\]

Now, let's perform the calculations:

1. For point \( A(4, 2) \):
\[
A' = \left( \frac{1}{2} \times 4, \frac{1}{2} \times 2 \right) = (2, 1)
\]

2. For point \( B(6, 6) \):
\[
B' = \left( \frac{1}{2} \times 6, \frac{1}{2} \times 6 \right) = (3, 3)
\]

Thus, after dilation, the coordinates are:
- \( A' = (2, 1) \)
- \( B' = (3, 3) \)

Now, you can check against the given options:

1. A′ is at (1,5) and B′ is at (3,6).
2. A′ is at (4,8) and B′ is at (12,12).
3. A′ is at (−1,−2) and B′ is at (−3,−3).
4. A′ is at (1,2) and B′ is at (3,3).

None of these options represent \( (2, 1) \) and \( (3, 3) \), so it seems that there was a misunderstanding in the interpretation of the dilation or the available options.

The coordinates we found after dilation:
- \( A' = (2, 1) \)
- \( B' = (3, 3) \)

If you also want to check the lengths as a validation step:
- The distance \( AB \) can be calculated using the distance formula:
\[
AB = \sqrt{(6 - 4)^2 + (6 - 2)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

- The distance \( A'B' \) is:
\[
A'B' = \sqrt{(3 - 2)^2 + (3 - 1)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\]

Clearly, \( A'B' \) is half of \( AB \), validating that the transformation correctly scaled \( AB \) by \( \frac{1}{2} \).
Answered by GPT-4o mini
Based on the provided information, here are the results:

After dilating points \( A(4, 2) \) and \( B(6, 6) \) by a scale factor of \( \frac{1}{2} \):

1. **Point \( A' \)**:
\[
A' = \left( \frac{1}{2} \times 4, \frac{1}{2} \times 2 \right) = (2, 1)
\]

2. **Point \( B' \)**:
\[
B' = \left( \frac{1}{2} \times 6, \frac{1}{2} \times 6 \right) = (3, 3)
\]

So, the coordinates after the dilation are:

- \( A' = (2, 1) \)
- \( B' = (3, 3) \)

### Answer:
None of the options from your original choices match \( A' = (2, 1) \) and \( B' = (3, 3) \).

If you're looking for a specific formatted answer from your original options:

- If the goal is to find \( A' \) and \( B' \), the correct answers would be:
- \( A' \) is at \( (2, 1) \)
- \( B' \) is at \( (3, 3) \)

However, these coordinates do not correspond to any of the options you provided. Please check to see if there are any alternative options or clarify if there's more context or information that might change the expected answer.

Related Questions