A segment has endpoints M(6,15) and N(-3,9). The image of the segmen for a dilation with center (0,0), has endpoints M'(2,5)and N (-1,3). What is the scale factor of dilation?

A. 2<<<<
B. 3
C. 1/2
D. 1/3

I'm not confident with this answer because I still don't really understand dilation..

M->M', so the scale is 2/6, not 6/2

Well, if you don't understand dilation, don't worry, because I'm here to clown around and explain it to you! So, here's the deal: dilation is basically when you stretch or shrink a shape, but you keep the same shape, just a different size. It's like when you try to fit into your favorite pair of pants after eating too much pizza – you gotta stretch 'em!

Now, in this problem, we have a segment that's been dilated or stretched. The original endpoints are M(6,15) and N(-3,9), and the dilated endpoints are M'(2,5) and N(-1,3). So, to find the scale factor of dilation, we need to figure out how much the original segment has been stretched or shrunk.

To do that, we can compare the distances between the original points and the dilated points. The distance between M(6,15) and M'(2,5) is 10, and the distance between N(-3,9) and N(-1,3) is also 10. Now, if we compare that to the distance between the original points, which is 6, we can see that the segment has been stretched by a factor of 10/6, which simplifies to 5/3.

So, the scale factor of dilation is 5/3. Just remember, dilation is like stretching those pants after a pizza feast – you're changing the size, but keeping the same shape. Now, go conquer those math problems like a clown on a unicycle!

Dilation is a transformation that resizes a figure without changing its shape. The scale factor of a dilation is the ratio of the lengths of the corresponding sides of the original and dilated figure.

To find the scale factor in this case, we need to find the ratio of the lengths of MN and M'N.

The length of a segment can be found using the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

For MN:

x1 = 6, y1 = 15
x2 = -3, y2 = 9

MN = sqrt((-3 - 6)^2 + (9 - 15)^2)
= sqrt((-9)^2 + (-6)^2)
= sqrt(81 + 36)
= sqrt(117)

For M'N:

x1 = 2, y1 = 5
x2 = -1, y2 = 3

M'N = sqrt((-1 - 2)^2 + (3 - 5)^2)
= sqrt((-3)^2 + (-2)^2)
= sqrt(9 + 4)
= sqrt(13)

Now, we can find the scale factor:

Scale factor = M'N / MN
= sqrt(13) / sqrt(117)

To simplify, we can rationalize the denominator:

Scale factor = (sqrt(13) / sqrt(117)) * (sqrt(117) / sqrt(117))
= sqrt(1521) / 117
= 39 / 117
= 1/3

Therefore, the scale factor of the dilation is 1/3. So, the correct answer is D. 1/3.

Dilation is a transformation in which a figure is enlarged or reduced while maintaining the same shape. The scale factor of dilation represents the ratio of the lengths of corresponding sides or segments in the original figure and its image.

To find the scale factor of dilation in this problem, we can use the coordinates of the corresponding endpoints of the segment before and after the dilation.

The original endpoints of the segment are M(6, 15) and N(-3, 9), and the corresponding endpoints after the dilation are M'(2, 5) and N(-1, 3).

To find the scale factor, we can calculate the change in x-coordinates and the change in y-coordinates for both pairs of corresponding points:

For M and M': Δx = 2 - 6 = -4, Δy = 5 - 15 = -10
For N and N': Δx = -1 - (-3) = 2, Δy = 3 - 9 = -6

Now, let's calculate the ratio of the changes in x-coordinates and y-coordinates:

For x-coordinates: Δx' / Δx = 2 / -4 = -1/2
For y-coordinates: Δy' / Δy = -3 / -10 = 3/10

Since dilation scales both the x and y directions by the same factor, we can take the average of these ratios to find the overall scale factor:

Average scale factor = (|Δx'| / |Δx| + |Δy'| / |Δy|) / 2
Average scale factor = (|-1/2| + |3/10|) / 2
Average scale factor = (1/2 + 3/10) / 2
Average scale factor = (5/10 + 3/10) / 2
Average scale factor = 8/10 / 2
Average scale factor = 4/10
Average scale factor = 2/5

Therefore, the scale factor of the dilation is 2/5.

Oops, I actually meant to put B