Q. 454.5( 6 Votes )
If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.
Our given vertices are A(1, -2), B(3, 6) and C(5, 10) and fourth vertex be D(k, p)
It is given that quadrilateral joining these four vertices is parallelogram, ie □ABCD is parallelogram.
We know that diagonals of parallelogram bisect each other, ie midpoint of the diagonals coincide.
Let E(xm , ym) be the midpoint of diagonals AC and BD.
By midpoint formula,
x = , y =
For diagonal AC,
xm = , ym =
∴ xm = , ym =
∴ E(xm , ym) ≡ (, )
For diagonal BD,
= , =
∴ k = 15 – 8 , y = 5 – 2
∴ k = 7 and p = 3
Hence, our fourth vertex is D(7 , 3)
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