Question

Use the given endpoint Y and the midpoint M of YZ (there is a line over YZ) to find the coordinates of the other endpoint Z

Answer 1

The x-coordinate of our midpoint = 3

The x-coordinate of Y = 0

k = x-coordinate of Z

3 = (0 + k) / 2

Multiply both sides by 2

6 = k + 0

Simplify

6 = k

The y-coordinate of our midpoint = 3

the y-coordinate of Y = 5

w = y-coordinate of Z

3 = (w + 5) / 2

Multiply both sides by 2

6 = w + 5

Subtract 5 from both sides

1 = w

For problem #17, Z(6, 1)

Let's do problem 18 now.

The x-coordinate of our midpoint = 5

The x-coordinate of Y = -1

k = x-coordinate of Z

5 = (k + (-1)) / 2

Multiply both sides by 2

10 = k + (-1)

Simplify the right side

10 = k - 1

Add 1 to both sides

11 = k

The y-coordinate of our midpoint = 9

The y-coordinate of Y = -3

w = y-coordinate of Z

9 = (w + (-3)) / 2

Multiply both sides by 2

18 = w + (-3)

Simplify the right side

18 = w - 3

Add 3 to both sides

21 = w

For problem #18, Z(11, 21)

Answer 2

The coordinates for point Y are (2, -5) and the coordinates for the midpoint M are (6, 3).

From these coordinates, we know that the midpoint M is the average of the coordinates for points Y and Z. The formula for finding a midpoint is M = [(x1+x2)/2, (y1+y2)/2].

However, in this case we want to go the other direction to find the x and y coordinates for point Z. So, we can rearrange the midpoint formula to solve for Z, yielding Z = [2Mx - Yx, 2My - Yy].

Let's plug in the given coordinates for Y and M:

For the X value of Z: 2*6 - 2 = 10,
For the Y value of Z: 2*3 - (-5) = 11.

Thus, the coordinates of point Z are (10,11).