Q. 74.4( 7 Votes )
If (x1, y1), (x2, y2), (x3, y3) and (x4, y4) point s are joined in order to form a parallelogram, then prove that x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4.
A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4)
We know that a quadrilateral is a parallelogram if the co-ordinates of mid-point s of its both the diagonals are same.
Therefore, we’ll find the mid-point s of diagonal AC and BD.
Let the co-ordinates of mid-point of AC be (x0, y0).
And, we know mid-point formula, i.e. the coordinates of mid-point of line joining (x1, y1) and (x2, y2) is
Similarly, let the co-ordinates of mid-point of AC be (x5, y5).
And, since it is a mid-point –
Now, since ABCD is a parallelogram –
⇒ (x0, y0) = (x5, y5)
⇒ x0 = x5 and y0 = y5
⇒x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4
Rate this question :
The point s P and Q lie on 1st and 3rd quadrant respectively. The distances of the two points from x – axis and y – axis are 6 units and 4 units respectively. Let us write the co-ordinates of mid-point of line segment PQ.West Bengal Mathematics
The co-ordinates of vertices of A, B, C of a triangle ABC are (–1, 3), (1, –1) and (5, 1) respectively, let us calculate the length of Median AD.West Bengal Mathematics
The co-ordinates of end point s of a diameter of a circle are (7, 9) and (–1, –3). The co-ordinates of centre of circle isWest Bengal Mathematics
The point s A and B lie on 2nd and 4th quadrant respectively and distance of each point from x – axis and y – axis are 8 units and 6 units respectively. Let us write the co – ordinate of mid-point of line segment AB.West Bengal Mathematics
The point P lies on the line segment AB and AP = PB; the co-ordinates of A and B are (3, –4) and (–5, 2) respectively. Let us write the co-ordinates of point ?West Bengal Mathematics
The co-ordinates of mid-point s of sides of a triangle are (4, 3), (–2, 7) and (0, 11). Let us calculate the co-ordinates of its vertices.West Bengal Mathematics
If the point s P(1, 2), Q(4, 6), R(5, 7) and S(x, y) are the vertices of a parallelogram PQRS, thenWest Bengal Mathematics
The co-ordinates of vertices of triangle are (2, –4), (6, –2) and (–4, 2) respectively. Let us find the length of three medians of triangle.West Bengal Mathematics
Find the co-ordinates of mid-point of line segment joining two point s for the following
(i) (5, 4) and (3, –4)
(ii) (6, 0) and (0, 7)West Bengal Mathematics