in triangle GHJ, K(2,3) is the midpoint of GH, L(4,1) is the midpoint of HJ, M(6,2) is the midpoint of GJ. Find the coordinates of G, H, and J.

G(0,5), H(2,3), J(4,1)

To find the coordinates of G, H, and J, we can use the properties of midpoints.

1. Let's start by finding the coordinates of G:
To find the coordinates of G, we'll use the midpoint formula:
The x-coordinate of G = 2 * x-coordinate of K - x-coordinate of M
= 2 * 2 - 6
= 4 - 6
= -2

The y-coordinate of G = 2 * y-coordinate of K - y-coordinate of M
= 2 * 3 - 2
= 6 - 2
= 4

So G = (-2, 4)

2. Now let's find the coordinates of H:
To find the coordinates of H, we'll use the midpoint formula:
The x-coordinate of H = 2 * x-coordinate of K - x-coordinate of L
= 2 * 2 - 4
= 4 - 4
= 0

The y-coordinate of H = 2 * y-coordinate of K - y-coordinate of L
= 2 * 3 - 1
= 6 - 1
= 5

So H = (0, 5)

3. Finally, let's find the coordinates of J:
To find the coordinates of J, we'll use the midpoint formula:
The x-coordinate of J = 2 * x-coordinate of L - x-coordinate of M
= 2 * 4 - 6
= 8 - 6
= 2

The y-coordinate of J = 2 * y-coordinate of L - y-coordinate of M
= 2 * 1 - 2
= 2 - 2
= 0

So J = (2, 0)

Therefore, the coordinates of G, H, and J are G = (-2, 4), H = (0, 5), and J = (2, 0).

To find the coordinates of points G, H, and J in triangle GHJ, we can use the given information about the midpoints.

Let's start with point K, which is the midpoint of GH. The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) is the average of their x-coordinates and the average of their y-coordinates. In other words, the coordinates of K can be found by taking the average of the x-coordinates and the average of the y-coordinates of G and H.

Given that K(2,3) is the midpoint of GH, we can set up the following equations:

xK = (xG + xH) / 2,
yK = (yG + yH) / 2.

Substituting the coordinates of K into these equations, we get:

2 = (xG + xH) / 2,
3 = (yG + yH) / 2.

Next, let's consider point L, which is the midpoint of HJ. Using the midpoint formula again, we have:

xL = (xH + xJ) / 2,
yL = (yH + yJ) / 2.

Substituting the coordinates of L, we get:

4 = (xH + xJ) / 2,
1 = (yH + yJ) / 2.

Finally, let's look at point M, which is the midpoint of GJ. Applying the midpoint formula, we get:

xM = (xG + xJ) / 2,
yM = (yG + yJ) / 2.

Substituting the coordinates of M, we have:

6 = (xG + xJ) / 2,
2 = (yG + yJ) / 2.

Now we have a system of equations with three unknowns (xG, xH, xJ, yG, yH, yJ) and three equations:

2 = (xG + xH) / 2,
3 = (yG + yH) / 2,
4 = (xH + xJ) / 2,
1 = (yH + yJ) / 2,
6 = (xG + xJ) / 2,
2 = (yG + yJ) / 2.

Solving this system of equations will give us the coordinates of G, H, and J.