Advertisements
Advertisements
प्रश्न
Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.
Advertisements
उत्तर
Let A (1, 0); B (5, 3); C (2, 7) and D (-2, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a parallelogram.
We should proceed with the fact that if the diagonals of a quadrilateral bisect each other than the quadrilateral is a parallelogram.
Now to find the mid-point P(x,y) of two points `A(x_1,y_1)`and `B(x_2, y_2)` we use section formula as,
`P(x,y) = ((x_1 + x_2)/2,(y_1 + y_2)/2)`
So the mid-point of the diagonal AC is,
`Q(x,y) = ((1 + 2)/2, (0 + 7)/2)`
`= (3/2, 7/2)`
Similarly mid-point of diagonal BD is,
`R(x,y) = ((5 - 2)/2, (3 + 4)/2)`
`= (3/2, 7/2)`
Therefore the mid-points of the diagonals are coinciding and thus diagonal bisects each other.
Hence ABCD is a parallelogram.
APPEARS IN
संबंधित प्रश्न
How will you describe the position of a table lamp on your study table to another person?
If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k
Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(-3, 5) B(3, 1), C (0, 3), D(-1, -4)
The line segment joining the points P(3, 3) and Q(6, -6) is trisected at the points A and B such that Ais nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.
Points A(-1, y) and B(5,7) lie on the circle with centre O(2, -3y).Find the value of y.
If the points A (2,3), B (4,k ) and C (6,-3) are collinear, find the value of k.
The ordinate of any point on x-axis is
If P ( 9a -2 , - b) divides the line segment joining A (3a + 1 , - 3 ) and B (8a, 5) in the ratio 3 : 1 , find the values of a and b .
ABCD is a parallelogram with vertices \[A ( x_1 , y_1 ), B \left( x_2 , y_2 \right), C ( x_3 , y_3 )\] . Find the coordinates of the fourth vertex D in terms of \[x_1 , x_2 , x_3 , y_1 , y_2 \text{ and } y_3\]
In \[∆\] ABC , the coordinates of vertex A are (0, - 1) and D (1,0) and E(0,10) respectively the mid-points of the sides AB and AC . If F is the mid-points of the side BC , find the area of \[∆\] DEF.
If (x, y) be on the line joining the two points (1, −3) and (−4, 2) , prove that x + y + 2= 0.
Find the value of a for which the area of the triangle formed by the points A(a, 2a), B(−2, 6) and C(3, 1) is 10 square units.
Find the value(s) of k for which the points (3k − 1, k − 2), (k, k − 7) and (k − 1, −k − 2) are collinear.
If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.
If x is a positive integer such that the distance between points P (x, 2) and Q (3, −6) is 10 units, then x =
The line 3x + y – 9 = 0 divides the line joining the points (1, 3) and (2, 7) internally in the ratio ______.
Ordinate of all points on the x-axis is ______.
(–1, 7) is a point in the II quadrant.
