Question 291653: let A(-3,2) , B(-1,-3) and C(3,4) be the three vertices of a parallelogram, determine the coordinate of the fourth point.
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! let A(-3,2) , B(-1,-3) and C(3,4) be the three vertices of a parallelogram, determine the coordinate of the fourth point.
A(-3,2)
B(-1,-3)
C(3,4)
see what we got on a grid:
D(?,?), length DC = ?, length AD = ?
AB and DC need to be equal length and parallel
BC and AD need to be equal length and parallel
parallel lines have the same slope
AB = sqrt((-1 - -3)^2 + (-3 - 2)^2)
AB = sqrt(2^2 + -5^2)
AB = sqrt(4 + 25) = sqrt(29) needs to equal DC
BC = sqrt((3 - -1)^2 + (4 - -3)^2)
BC = sqrt(4^2 + 7^2)
BC = sqrt(16 + 49) = sqrt(65) needs to equal AD
slope AB = (-3 - 2)/(-1 - -3) = -5/2 needs to be same as slope DC
slope BC = (4 - -3)/(3 - -1) = 7/4 needs to be same as slope AD
call x and y the x and y coordinates of D
AB = DC = sqrt(29) = sqrt((x - 3)^2 + (y - 4)^2)
29 = x^2 - 6x + 9 + y^2 - 8y + 16
slope AB = slope DC = -5/2 = (y - 4)/(x - 3)
-5(x - 3) = 2(y - 4)
-5x + 15 = 2y - 8
-5x - 2y = -23
-5x + 23 = 2y
(-5/2)x + (23/2) = y
BC = AD = sqrt(65) = sqrt((x + 3)^2 + (y - 2)^2)
65 = x^2 + 6x + 9 + y^2 - 4y + 4
slope BC = slope AD = 7/4 = (y - 2)/(x + 3)
7(x + 3) = 4(y - 2)
7x + 21 = 4y - 8
7x - 4y = -29
7x + 29 = 4y
(7/4)x + (29/4) = y
system of equations:
-5x - 2y = -23 (multiply by -2 and add to 2nd) --> 10x + 4y = 46
7x - 4y = -29
--> -5x - 2y = -23
17x + 0y = 17
x = 1 --> -5 - 2y = -23 --> -2y = -18 --> y = 9
so D is (1,9)
checking: slope AB = slope DC = -5/2 = (y - 4)/(x - 3) = 5/(-2) = -5/2
slope BC = slope AD = 7/4 = (y - 2)/(x + 3) = 7/4
AB = DC = sqrt(29) = sqrt((x - 3)^2 + (y - 4)^2) =
= sqrt(-2^2 + 5^2)
= sqrt(4 + 25) = sqrt(29)
BC = AD = sqrt(65) = sqrt((x + 3)^2 + (y - 2)^2) =
= sqrt(4^2 + 7^2)
= sqrt(16 + 49) = sqrt(65)
so D is indeed (1,9)
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