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Sol.
Mid-point of (x 1 , y 1 ) and (x 2 , y 2 ) ,

Mid point of AC is (1, 2)

Mid point of BD is (1, 2)
Since, diagonals AC and BD have the same mid-points, therefore, AC and BD
bisect each other. Hence, ABCD is a parallelogram.

Exa:13- If a vertex of a triangle be (1, 1) and the mid-points of the sides through
it be (-2, 3) and (5, 2). Find the other vertices.
Sol.

Let A(1, 1) be co-ordinates of the vertex A and (-2, 3) and (5, 2) be the co-
ordinates of the mid-points of AB and AC.
Let co-ordinates of B and C be (x 1 , y 1 ) and (x 2 , y 2 ) respectively.
Using mid-point formula for side AB

x 1 + 1 = -4
x 1 = -5

y 1 + 1 = 6
y 1 = 5
Similarly, using mid-point formula for the side AC.

x 2 + 1 = 10
x 2 = 9

y 2 + 1 = 4
y 2 = 3
Thus, co-ordinates of B and C are (-5, 5) and (9, 3) respectively.

Exa:14- Calculate the ratio in which the line joining A (6, 5) and B (4, - 3) is
divided by the line y = 2.

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