Answer:
the scale factor (k) for the dilation that maps segment AB to A'B' is 3/4. This means that the image segment A'B' is 3/4 times the length of the original segment AB.
Explanation:
To find the scale factor for the dilation that maps segment AB to A'B', we can use the following formula:
Scale factor (k) = Distance of image segment / Distance of original segment
Step 1: Find the distance between points A and B using the distance formula:
Distance between A and B = √[(x2 - x1)^2 + (y2 - y1)^2]
Plugging in the coordinates of points A and B:
x1 = 9, y1 = 4
x2 = 5, y2 = -4
Distance between A and B = √[(5 - 9)^2 + (-4 - 4)^2]
= √[(-4)^2 + (-8)^2]
= √[16 + 64]
= √80
Step 2: Find the distance between points A' and B' using the distance formula:
Distance between A' and B' = √[(x2' - x1')^2 + (y2' - y1')^2]
Plugging in the coordinates of points A' and B':
x1' = 6, y1' = 3
x2' = 3, y2' = -3
Distance between A' and B' = √[(3 - 6)^2 + (-3 - 3)^2]
= √[(-3)^2 + (-6)^2]
= √[9 + 36]
= √45
Step 3: Calculate the scale factor (k) by dividing the distance of the image segment by the distance of the original segment:
Scale factor (k) = Distance of image segment / Distance of original segment
= √45 / √80
Step 4: Simplify the scale factor (k):
Scale factor (k) = √45 / √80
= √(9/5) / √(16/5)
= (3/√5) / (4/√5)
= (3/√5) * (√5/4)
= 3/4
So, the scale factor (k) for the dilation that maps segment AB to A'B' is 3/4. This means that the image segment A'B' is 3/4 times the length of the original segment AB.