Answer:
Explanation:
To find the coordinates of endpoint K and the distance between the endpoints of JK, we can use the midpoint formula and the distance formula.
Midpoint Formula:
The midpoint formula states that the coordinates of the midpoint (M) of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
Given that the midpoint M is (2, 5) and one endpoint is J (-4, 2), we can use the midpoint formula to find K:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
(2, 5) = ((-4 + x2) / 2, (2 + y2) / 2)
Now, we can solve for x2 and y2:
2 = (-4 + x2) / 2
5 = (2 + y2) / 2
First, solve for x2:
2 = (-4 + x2) / 2
2 * 2 = -4 + x2
4 = -4 + x2
Now, add 4 to both sides to isolate x2:
4 + 4 = -4 + 4 + x2
8 = x2
So, the x-coordinate of endpoint K is 8.
Next, solve for y2:
5 = (2 + y2) / 2
5 * 2 = 2 + y2
10 = 2 + y2
Now, subtract 2 from both sides to isolate y2:
10 - 2 = 2 - 2 + y2
8 = y2
So, the y-coordinate of endpoint K is 8.
Distance Formula:
To find the distance between the endpoints J and K, you can use the distance formula, which is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substitute the coordinates of J and K into the formula:
Distance = √((8 - (-4))^2 + (8 - 2)^2)
Distance = √((8 + 4)^2 + (8 - 2)^2)
Distance = √(12^2 + 6^2)
Distance = √(144 + 36)
Distance = √180
Distance = 6√5
So, the distance between the endpoints J and K is 6√5 units.