Asked by bot

AC¯¯¯¯¯¯¯¯ is a dilation of AB¯¯¯¯¯¯¯¯ , with point B as the center of dilation. Determine the scale factor. (1 point)
point A (1,-1)
point B (2,2)
point C (5,5)
Answered by bot
So what's the scale factor
Answered by bot
I don't understand what that means can you tell me the answer without using / (?
Answered by bot
Thank you so much this is very helpful
Answered by GPT-4o mini
To determine the scale factor of the dilation \( AC \) with respect to the center of dilation \( B \), we can use the formula for the scale factor \( k \) given by the ratio of the distances:

\[
k = \frac{\text{distance from } B \text{ to } C}{\text{distance from } B \text{ to } A}
\]

First, we need to find the distances from point \( B \) to points \( A \) and \( C \).

1. **Calculate the distance from \( B \) to \( A \)**:

Point \( A = (1, -1) \) and point \( B = (2, 2) \).

\[
\text{Distance } d(B, A) = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\]
\[
d(B, A) = \sqrt{(2 - 1)^2 + (2 - (-1))^2}
\]
\[
d(B, A) = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

2. **Calculate the distance from \( B \) to \( C \)**:

Point \( C = (5, 5) \) and point \( B = (2, 2) \).

\[
d(B, C) = \sqrt{(x_B - x_C)^2 + (y_B - y_C)^2}
\]
\[
d(B, C) = \sqrt{(2 - 5)^2 + (2 - 5)^2}
\]
\[
d(B, C) = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]

3. **Now calculate the scale factor \( k \)**:

\[
k = \frac{d(B, C)}{d(B, A)} = \frac{3\sqrt{2}}{\sqrt{10}}
\]

4. **Simplify the scale factor**:

To simplify, we can multiply the numerator and the denominator by \( \sqrt{10} \):

\[
k = \frac{3\sqrt{2}}{\sqrt{10}} = \frac{3\sqrt{2} \cdot \sqrt{10}}{10} = \frac{3\sqrt{20}}{10} = \frac{3 \cdot 2\sqrt{5}}{10} = \frac{6\sqrt{5}}{10} = \frac{3\sqrt{5}}{5}
\]

Thus, the scale factor \( k \) of the dilation \( AC \) with respect to point \( B \) is:

\[
\boxed{\frac{3\sqrt{5}}{5}}
\]
Answered by GPT-4o mini
The scale factor of the dilation \( AC \) with respect to point \( B \) is

\[
\frac{3\sqrt{2}}{\sqrt{10}}.
\]

Upon further simplification, the scale factor can also be represented as

\[
\frac{3\sqrt{5}}{5}.
\]

Thus, the scale factor is

\[
\boxed{\frac{3\sqrt{2}}{\sqrt{10}}} \quad \text{or} \quad \boxed{\frac{3\sqrt{5}}{5}}.
\]

Both forms represent the same scale factor. You can choose either depending on the context or preference.
Answered by GPT-4o mini
Absolutely! The scale factor can be given in decimal form instead of a fraction.

Let's find the decimal value of the scale factor \( \frac{3\sqrt{2}}{\sqrt{10}} \):

1. Calculate \( \sqrt{2} \) and \( \sqrt{10} \):
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{10} \approx 3.162 \)

2. Plug these values into the scale factor:

\[
k \approx \frac{3 \times 1.414}{3.162} \approx \frac{4.242}{3.162} \approx 1.34
\]

So in decimal form, the scale factor is approximately **1.34**.

Therefore, the scale factor is approximately

\[
\boxed{1.34}
\]

If that’s what you prefer!
Answered by GPT-4o mini
You're very welcome! I'm glad I could help. If you have any more questions or need further clarification on anything else, feel free to ask. Happy studying!