What is the scale factor of a dilation centered at the origin that maps the point (12, -8) to the point (-7.2, 4.8)?
looks like ... -.6
what is the factor of a dilation centered at the origin that maps the point (10,-4) to the point (-5,2)
To find the scale factor of a dilation, we can use the formula:
scale factor = distance after dilation / distance before dilation
In this case, we need to find the distance from the origin to the points (12, -8) and (-7.2, 4.8).
The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
For the distance before dilation (from the origin to (12, -8)):
x1 = 0, y1 = 0 (origin)
x2 = 12, y2 = -8
d_before = √((12 - 0)^2 + (-8 - 0)^2)
= √(12^2 + (-8)^2)
= √(144 + 64)
= √208
≈ 14.42
For the distance after dilation (from the origin to (-7.2, 4.8)):
x1 = 0, y1 = 0 (origin)
x2 = -7.2, y2 = 4.8
d_after = √((-7.2 - 0)^2 + (4.8 - 0)^2)
= √((-7.2)^2 + 4.8^2)
= √(51.84 + 23.04)
= √74.88
≈ 8.66
Now we can calculate the scale factor:
scale factor = distance after dilation / distance before dilation
= d_after / d_before
= 8.66 / 14.42
≈ 0.6
Therefore, the scale factor of the dilation centered at the origin that maps the point (12, -8) to the point (-7.2, 4.8) is approximately 0.6.
To find the scale factor of a dilation, we can use the formula:
Scale factor = Distance of image point / Distance of pre-image point
In this case, the distance between the pre-image point (12, -8) and the origin is √(12² + (-8)²). We can find the distance using the Pythagorean theorem.
Distance = √(12² + (-8)²) = √(144 + 64) = √208
Similarly, the distance between the image point (-7.2, 4.8) and the origin is √((-7.2)² + 4.8²).
Distance = √((-7.2)² + 4.8²) = √(51.84 + 23.04) = √(74.88)
Now we can use the formula to find the scale factor:
Scale factor = Distance of image point / Distance of pre-image point
= √(74.88) / √208
Simplifying this expression, we get:
Scale factor = √74.88 / √208
= √(74.88/208)
= √0.36
= 0.6
Therefore, the scale factor of the dilation is 0.6.