A segment with endpoints (3, -2) and (4, 2) is dilated to the image segment with endpoints (9, -6) and (12, 6). What is the scale factor for the dilation?
3
2
6 **my choice
9 **i feel its this one too, im stuck
Can you just answer the freaking question?
To find the scale factor for the dilation, we can compare the distances between corresponding points on the original segment to the distances between corresponding points on the dilated segment.
The distance between the two points (3, -2) and (4, 2) can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Applying the formula, we get:
d = sqrt((4 - 3)^2 + (2 - (-2))^2)
= sqrt(1^2 + 4^2)
= sqrt(1 + 16)
= sqrt(17)
Similarly, the distance between the two points (9, -6) and (12, 6) can be calculated:
d' = sqrt((12 - 9)^2 + (6 - (-6))^2)
= sqrt(3^2 + 12^2)
= sqrt(9 + 144)
= sqrt(153)
The scale factor, denoted as k, can be found by dividing the distance between the corresponding points on the dilated segment (d') by the distance on the original segment (d):
k = d' / d
= (sqrt(153)) / (sqrt(17))
Therefore, the scale factor for this dilation is the square root of 153 divided by the square root of 17. None of the options provided match this result.
To find the scale factor for the dilation, we can compare the lengths of the original segment and the dilated segment.
The length of a segment can be found using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2). So, for the original segment with endpoints (3, -2) and (4, 2), the length is:
√((4 - 3)^2 + (2 - (-2))^2) = √(1^2 + 4^2) = √(1 + 16) = √17.
Similarly, for the dilated segment with endpoints (9, -6) and (12, 6), the length is:
√((12 - 9)^2 + (6 - (-6))^2) = √(3^2 + 12^2) = √(9 + 144) = √153.
Now, to find the scale factor, we need to divide the length of the dilated segment by the length of the original segment:
√153 / √17.
Simplifying this, we can multiply both the numerator and denominator by √17:
(√153 * √17) / (√17 * √17).
This gives us:
√(153 * 17) / 17 = √2601 / 17 = 51 / 17 = 3.
Therefore, the scale factor for the dilation is 3.
Oh my goodness!
What has happened to each of the coordiates of the two points?
Why was 6 your choice?
How can you "feel" an answer to be right?