After a dilation centered at the origin, the image of AB is
A’B’. If the coordinates of the endpoints of these segments are
A (6,-4),
B (2,-8)
A’ (9,-6), and
B’ (3,-12), what is the scale factor of the dilation?
1. 3/2
2. 1/2
3. 2/3
4. 3
9/6 = 3/2
check the other number pairs to be sure they all agree
To find the scale factor of the dilation, we can use the formula:
scale factor = length of image segment / length of pre-image segment
The length of AB can be found using the distance formula:
length of AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
Substituting the given coordinates, we have:
length of AB = √[(2 - 6)² + (-8 - (-4))²] = √[(-4)² + (-4)²] = √[16 + 16] = √32
The length of A'B' can also be found using the distance formula:
length of A'B' = √[(x₂ - x₁)² + (y₂ - y₁)²]
Substituting the given coordinates, we have:
length of A'B' = √[(3 - 9)² + (-12 - (-6))²] = √[(-6)² + (-6)²] = √[36 + 36] = √72
Now, we can calculate the scale factor:
scale factor = length of A'B' / length of AB
scale factor = √72 / √32
Simplifying the radical expressions, we have:
scale factor = √(9 * 8) / √(4 * 8) = 3√2 / 2√2 = 3/2
Therefore, the scale factor of the dilation is 3/2, which corresponds to option 1.
To find the scale factor of the dilation, we need to compare the lengths of the original line segment AB and the image line segment A'B'.
The formula to calculate the distance between two points is the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the length of AB using the distance formula:
d_AB = sqrt((2 - 6)^2 + (-8 - (-4))^2)
= sqrt((-4)^2 + (-4)^2)
= sqrt(16 + 16)
= sqrt(32)
Now, let's calculate the length of A'B':
d_A'B' = sqrt((3 - 9)^2 + (-12 - (-6))^2)
= sqrt((-6)^2 + (-6)^2)
= sqrt(36 + 36)
= sqrt(72)
The scale factor of the dilation is given by the ratio of the lengths:
scale factor = d_A'B' / d_AB
= sqrt(72) / sqrt(32)
To simplify the expression, we can factor out the common factor of 2 from both the numerator and denominator:
scale factor = (2 * sqrt(18)) / (2 * sqrt(8))
Now, simplify further by dividing both the numerator and denominator by sqrt(8):
scale factor = sqrt(18) / sqrt(8)
= sqrt(9 * 2) / sqrt(4 * 2)
= sqrt(9) / sqrt(4) (since sqrt(2) / sqrt(2) = 1)
= 3/2
Therefore, the scale factor of the dilation is 3/2.