in triangle GHJ, k(2,3) is the midpoint of GH, l(4,1) is the midpoint of HJ and m(6,2) is the midpoint of GJ. Find the coordinates of G,H and J.
can you please show me how to find this? i don't really understand how it would be done.
thank you.
To find the coordinates of G, H, and J, we can first use the information given to find the coordinates of K, L, and M. Then, based on the relationships between the midpoints and the vertices of the triangle, we can find the coordinates of G, H, and J.
Given information:
K(2,3) is the midpoint of GH.
L(4,1) is the midpoint of HJ.
M(6,2) is the midpoint of GJ.
Step 1: Finding the coordinates of K, L, and M
The midpoint formula states that the coordinates of a midpoint between two points A(x1, y1) and B(x2, y2) can be found by taking the average of their respective x-coordinates and y-coordinates.
So, to find K, we can use the midpoint formula with the coordinates of G(x, y) and K(2, 3):
(x + 2)/2 = 2 => x + 2 = 4 => x = 4 - 2 => x = 2
(y + 3)/2 = 3 => y + 3 = 6 => y = 6 - 3 => y = 3
So, K is located at (2, 3).
Similarly, to find L, we can use the midpoint formula with the coordinates of H(x, y) and L(4, 1):
(x + 4)/2 = 4 => x + 4 = 8 => x = 8 - 4 => x = 4
(y + 1)/2 = 1 => y + 1 = 2 => y = 2 - 1 => y = 1
So, L is located at (4, 1).
To find M, we can use the midpoint formula with the coordinates of G(x, y) and M(6, 2):
(x + 6)/2 = 6 => x + 6 = 12 => x = 12 - 6 => x = 6
(y + 2)/2 = 2 => y + 2 = 4 => y = 4 - 2 => y = 2
So, M is located at (6, 2).
Step 2: Finding the coordinates of G, H, and J.
Now that we have the coordinates of K, L, and M, we can use the given relationships to find the coordinates of the vertices of the triangle.
K is the midpoint of GH, so the average of the x-coordinates of G and H should equal the x-coordinate of K. Similarly, the average of the y-coordinates of G and H should equal the y-coordinate of K.
(x + 2)/2 = 2 => x + 2 = 4 => x = 4 - 2 => x = 2
(y + y)/2 = 3 => 2y = 6 => y = 6/2 => y = 3
So, G is located at (2, 3).
L is the midpoint of HJ, so the average of the x-coordinates of H and J should equal the x-coordinate of L. Similarly, the average of the y-coordinates of H and J should equal the y-coordinate of L.
(x + 4)/2 = 4 => x + 4 = 8 => x = 8 - 4 => x = 4
(y + 1)/2 = 1 => y + 1 = 2 => y = 2 - 1 => y = 1
So, H is located at (4, 1).
M is the midpoint of GJ, so the average of the x-coordinates of G and J should equal the x-coordinate of M. Similarly, the average of the y-coordinates of G and J should equal the y-coordinate of M.
(2 + x)/2 = 6 => 2 + x = 12 => x = 12 - 2 => x = 10
(3 + y)/2 = 2 => 3 + y = 4 => y = 4 - 3 => y = 1
So, J is located at (10, 1).
Therefore, G is located at (2, 3), H is located at (4, 1), and J is located at (10, 1).
I hope this explanation helps you understand how to find the coordinates of the vertices in a triangle using midpoints! Let me know if you have any further questions.