https://file.aiquickdraw.com/mathbot/file/1727231617259vouhnlf4.jpg

The problem in the image asks to determine the images of given points after dilation (scaling) from the origin \( O \) with the specified scaling factor. Let's solve the problems step by step.

### Formula for Dilation
When dilating a point \( P(x, y) \) from the origin \( O \), the new point \( P'(x', y') \) can be found by multiplying both the x-coordinate and the y-coordinate by the scaling factor \( k \):
\[
(x', y') = (k \cdot x, k \cdot y)
\]
where \( k \) is the scaling factor.

### Let's solve each part:

**a. Point \( (3, 4) \) with scaling factor \( 2 \):**
\[
(3', 4') = (2 \cdot 3, 2 \cdot 4) = (6, 8)
\]
So, the new point is \( (6, 8) \).

**b. Point \( (-2, 4) \) with scaling factor \( -4 \):**
\[
(-2', 4') = (-4 \cdot (-2), -4 \cdot 4) = (8, -16)
\]
So, the new point is \( (8, -16) \).

**c. Point \( (2, -1) \) with scaling factor \( 3 \):**
\[
(2', -1') = (3 \cdot 2, 3 \cdot (-1)) = (6, -3)
\]
So, the new point is \( (6, -3) \).

**d. Point \( (-6, 2) \) with scaling factor \( \frac{1}{2} \):**
\[
(-6', 2') = \left(\frac{1}{2} \cdot (-6), \frac{1}{2} \cdot 2\right) = (-3, 1)
\]
So, the new point is \( (-3, 1) \).

**e. Point \( (12, -27) \) with scaling factor \( \frac{1}{3} \):**
\[
(12', -27') = \left(\frac{1}{3} \cdot 12, \frac{1}{3} \cdot (-27)\right) = (4, -9)
\]
So, the new point is \( (4, -9) \).

### Final Answers:
- a. \( (6, 8) \)
- b. \( (8, -16) \)
- c. \( (6, -3) \)
- d. \( (-3, 1) \)
- e. \( (4, -9) \)

Would you like any further details on dilation or have any questions?

### Related questions:
1. How do you perform a dilation if the center of dilation is not the origin?
2. What happens to the orientation of a shape when dilated with a negative scaling factor?
3. Can dilation result in a smaller image? If yes, under what conditions?
4. How does dilation affect the distances between points on a geometric figure?
5. What are some real-world applications of dilation in mathematics?

**Tip:** When the scaling factor is less than 1, the image becomes smaller, while a factor greater than 1 enlarges the image.

Tentukanlah bayangan titik-titik berikut oleh dilatasi yang diberikan.

Solve the problem of finding the image of points after dilation with a given scaling factor from the origin. This problem includes points such as (3, 4) and (-2, 4), dilated by factors like 2 and -4. Suitable for Grade 9-10 students.

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