Final answer:
To find the length of each side of MNK, we use the midpoint formula to find the coordinates of M, N, and K. Then, we use the distance formula to find the length of each side of MNK. The length of MN is 5, NK is 3, and KM is 2.
Step-by-step explanation:
To find the length of each side of MNK, we need to find the midpoints of the sides of triangle ABC. The midpoint of AB is M, the midpoint of BC is N, and the midpoint of AC is K. To find the coordinates of M, N, and K, we use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
(x1 + x2)/2, (y1 + y2)/2
Using the given information AB = 8, BC = 10, and AC = 12, we can find the coordinates of M, N, and K:
M: ((x1 + x2)/2, (y1 + y2)/2) = ((0 + 8)/2, (0 + 0)/2) = (4, 0)
N: ((x1 + x2)/2, (y1 + y2)/2) = ((8 + 10)/2, (0 + 0)/2) = (9, 0)
K: ((x1 + x2)/2, (y1 + y2)/2) = ((0 + 12)/2, (0 + 0)/2) = (6, 0)
To find the length of each side of MNK, we use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of M, N, and K, we can find the length of each side of MNK:
MN: sqrt((9 - 4)^2 + (0 - 0)^2) = sqrt(5^2) = 5
NK: sqrt((6 - 9)^2 + (0 - 0)^2) = sqrt((-3)^2) = 3
KM: sqrt((6 - 4)^2 + (0 - 0)^2) = sqrt(2^2) = 2