Find the coordinates of the midpoint of a segment having the given endpoints. Then find the distance between the endpoints.

1. G(0,0), H(-3,4)
2. A(-4,-4), W(-2,2)
3. N(5,2), K(-2,8)

To find the coordinates of the midpoint of a segment with given endpoints, we can use the midpoint formula, which states that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.

To find the distance between the endpoints, we can use the distance formula, which is derived from the Pythagorean theorem. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by the formula:

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

Let's apply these formulas to solve the given problems:

1. G(0,0), H(-3,4):
- Midpoint:
x-coordinate of midpoint = (0 + (-3))/2 = -3/2
y-coordinate of midpoint = (0 + 4)/2 = 2/2 = 1
Therefore, the midpoint of the segment GH is (-3/2, 1).

- Distance:
d = sqrt((-3 - 0)^2 + (4 - 0)^2)
= sqrt((-3)^2 + 4^2)
= sqrt(9 + 16)
= sqrt(25)
= 5
Therefore, the distance between G(0,0) and H(-3,4) is 5 units.

2. A(-4,-4), W(-2,2):
- Midpoint:
x-coordinate of midpoint = (-4 + (-2))/2 = -6/2 = -3
y-coordinate of midpoint = (-4 + 2)/2 = -2/2 = -1
Therefore, the midpoint of the segment AW is (-3, -1).

- Distance:
d = sqrt((-2 - (-4))^2 + (2 - (-4))^2)
= sqrt(2^2 + 6^2)
= sqrt(4 + 36)
= sqrt(40)
Therefore, the distance between A(-4,-4) and W(-2,2) is sqrt(40) units.

3. N(5,2), K(-2,8):
- Midpoint:
x-coordinate of midpoint = (5 + (-2))/2 = 3/2 = 1.5
y-coordinate of midpoint = (2 + 8)/2 = 10/2 = 5
Therefore, the midpoint of the segment NK is (1.5, 5).

- Distance:
d = sqrt((-2 - 5)^2 + (8 - 2)^2)
= sqrt((-7)^2 + 6^2)
= sqrt(49 + 36)
= sqrt(85)
Therefore, the distance between N(5,2) and K(-2,8) is sqrt(85) units.

In summary, to find the coordinates of the midpoint of a segment, we average the x-coordinates of the endpoints to get the x-coordinate of the midpoint, and we average the y-coordinates of the endpoints to get the y-coordinate of the midpoint. To find the distance between two points, we use the distance formula, which involves taking the square root of the sum of the squares of the differences of the x-coordinates and y-coordinates of the two points.

I will do one. I can check your work on the others if you wish.

b. midpoint (-4+-2)/2, (-4-2)/2

distance: sqrt((-4--2)^2 + (-4-2)^2)
distance; sqrt (4+36)=sqrt40