What is the scale factor of this dilation if the points have went from (-1,1), (-2,-2), (2,-2), and (3,1) to (-3,3), (-6,-6), (6,-6), and (9,3)? Is this a reduction or enlargement?
To find the scale factor of the dilation, we can compare the distance between corresponding points.
The distance between the original points (-1,1) and (-2,-2) is:
√((-2 - (-1))^2 + (-2 - 1)^2) = √((-1)^2 + (-3)^2) = √(1 + 9) = √10
The distance between the new points (-3,3) and (-6,-6) is:
√((-6 - (-3))^2 + (-6 - 3)^2) = √((-3)^2 + (-9)^2) = √(9 + 81) = √90
The scale factor can be found by dividing the distance between the new points by the distance between the original points:
√90 / √10 = (√90 / √10) * (√10 / √10) = √900 / √100 = 30 / 10 = 3.
The scale factor is 3, which means this is an enlargement.
To find the scale factor of the dilation, we can compare the distances between corresponding points before and after the dilation. Let's calculate the distances:
Before dilation:
- Distance between (-1,1) and (-2,-2):
d₁ = √[(-2 - (-1))² + (-2 - 1)²]
= √[(-1)² + (-3)²]
= √[1 + 9]
= √10
- Distance between (-2,-2) and (2,-2):
d₂ = √[(2 - (-2))² + (-2 - (-2))²]
= √[(4)² + (0)²]
= √[16 + 0]
= √16
= 4
- Distance between (2,-2) and (3,1):
d₃ = √[(3 - 2)² + (1 - (-2))²]
= √[(1)² + (3)²]
= √[1 + 9]
= √10
After dilation:
- Distance between (-3,3) and (-6,-6):
D₁ = √[(-6 - (-3))² + (-6 - 3)²]
= √[(-3)² + (-9)²]
= √[9 + 81]
= √90
- Distance between (-6,-6) and (6,-6):
D₂ = √[(6 - (-6))² + (-6 - (-6))²]
= √[(12)² + (0)²]
= √[144 + 0]
= √144
= 12
- Distance between (6,-6) and (9,3):
D₃ = √[(9 - 6)² + (3 - (-6))²]
= √[(3)² + (9)²]
= √[9 + 81]
= √90
To find the scale factor, we can compare the corresponding distances before and after the dilation:
- Scale factor for d₁ to D₁: D₁/d₁ = (√90)/ (√10)
- Scale factor for d₂ to D₂: D₂/d₂ = 12/4 = 3
- Scale factor for d₃ to D₃: D₃/d₃ = (√90)/ (√10)
Since all the scale factors are equal, we can say that the scale factor is (√90)/ (√10) or 3.
Since the scale factor is greater than 1, this dilation is an enlargement.
To find the scale factor of a dilation, you need to determine the ratio of the corresponding side lengths or the distance between the corresponding points in both the original and image figures.
Let's start by identifying the corresponding points in the original figure and the image figure:
(-1, 1) corresponds to (-3, 3)
(-2, -2) corresponds to (-6, -6)
(2, -2) corresponds to (6, -6)
(3, 1) corresponds to (9, 3)
We can calculate the distance between the corresponding points in the original figure and the image figure to find the scale factor.
For the first pair of points, the distance in the original figure is:
√[(x₂ - x₁)² + (y₂ - y₁)²] = √[(-1 - (-2))² + (1 - (-2))²] = √[1 + 9] = √10
The distance in the image figure is:
√[(x₂' - x₁')² + (y₂' - y₁')²] = √[(-3 - (-6))² + (3 - (-6))²] = √[9 + 81] = √90
Now, let's calculate the ratio of the distances between the corresponding points:
Ratio = Distance in the image figure / Distance in the original figure = √90 / √10
Simplifying the ratio:
Ratio = (√90 / √10) * (√10 / √10) = √(90/10) = √9 = 3
The scale factor of this dilation is 3.
To determine whether it's a reduction or enlargement, we examine the scale factor.
If the scale factor is greater than 1, it is an enlargement.
If the scale factor is between 0 and 1, it is a reduction.
In this case, the scale factor is 3, which is greater than 1, so it is an enlargement.