Find the coordinates of the other endpoint of a segment with the given endpoit and midpoint M.

R(2,6)
M(-1,1)

Please show steps.

To find the coordinates of the other endpoint of the segment, we need to use the concept of symmetry. Since M is the midpoint of the segment, we can find the distance between R and M and then use it to find the other endpoint.

Here are the steps:

1. Calculate the distance between R and M. We can use the distance formula:

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

Substitute the coordinates of R(2,6) and M(-1,1) into the formula:

Distance = sqrt((-1 - 2)^2 + (1 - 6)^2)

Distance = sqrt((-3)^2 + (-5)^2)

Distance = sqrt(9 + 25)

Distance = sqrt(34)

2. Since M is the midpoint, the distance between R and M is the same as the distance between the other endpoint and M. Therefore, we need to double the distance:

Distance = 2 * sqrt(34)

3. Now, we need to find the direction of the other endpoint. To do this, subtract the x-coordinate of M from the x-coordinate of R, and subtract the y-coordinate of M from the y-coordinate of R:

Δx = 2 - (-1) = 3
Δy = 6 - 1 = 5

4. Multiply the differences by 2 (to double the distance):

Δx = 2 * 3 = 6
Δy = 2 * 5 = 10

5. Finally, add the doubled differences to the x-coordinate and y-coordinate of M to find the coordinates of the other endpoint:

Other Endpoint = (Mx + Δx, My + Δy)

Other Endpoint = (-1 + 6, 1 + 10)

Other Endpoint = (5, 11)

Therefore, the coordinates of the other endpoint are (5, 11).