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First, we can use the distance formula to find the length of LM:
d(L,M) = sqrt((4 - (-3))^2 + (9 - 1)^2) = sqrt(49 + 64) = sqrt(113)
Since LM:MN = 2:3, we can express the distance from L to N as (3/2) times the distance from L to M:
d(L,N) = (3/2) * d(L,M) = (3/2) * sqrt(113)
To find the coordinates of N, we need to determine the direction from M to N. We know that LMN is a straight line, so the direction from M to N is the same as the direction from L to M. We can find this direction by subtracting the coordinates of L from the coordinates of M:
direction = (4 - (-3), 9 - 1) = (7, 8)
To find the coordinates of N, we start at M and move in the direction of LMN for a distance of (3/2) * d(L,M):
N = M + (3/2) * d(L,M) * direction / ||direction||
where ||direction|| is the length of the direction vector, which is:
||direction|| = sqrt(7^2 + 8^2) = sqrt(113)
Substituting the values, we get:
N = (4, 9) + (3/2) * sqrt(113) * (7/sqrt(113), 8/sqrt(113))
Simplifying, we get:
N = (4 + (21/2), 9 + (24/2)) = (14.5, 21)
Therefore, the coordinates of N are (14.5, 21).
Answered by ChatGPT 3
What else can I help you with?
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