Advertisements
Advertisements
Question
Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.
Solution
Let A (4, 5); B (7, 6); C (6, 3) and D (3, 2) 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) = ((4 + 6)/2, (5 + 3)/2)`
= (5,4)
Therefore the mid-points of the diagonals are coinciding and thus diagonal bisects each other.
Hence ABCD is a parallelogram.
Now to check if ABCD is a rectangle, we should check the diagonal length.
`AC = sqrt((6 - 4)^2 + (3 - 5)^2)`
`= sqrt(4 + )4`
`= 2sqrt2`
Similarly,
`BD = sqrt((7 - 3)^2 + (6 - 2)^2)`
`= sqrt(16 + 16)`
`= 4sqrt2`
Diagonals are of different lengths.
Hence ABCD is not a rectangle.
APPEARS IN
RELATED QUESTIONS
On which axis do the following points lie?
R(−4,0)
Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when A coincides with the origin and AB and AD are along OX and OY respectively.
The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.
If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.
The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x - y + k = 0. Find the value of k.
The midpoint of the line segment joining A (2a, 4) and B (-2, 3b) is C (1, 2a+1). Find the values of a and b.
In what ratio does the line x - y - 2 = 0 divide the line segment joining the points A (3, 1) and B (8, 9)?
The midpoint P of the line segment joining points A(-10, 4) and B(-2, 0) lies on the line segment joining the points C(-9, -4) and D(-4, y). Find the ratio in which P divides CD. Also, find the value of y.
Find the coordinates of the centre of the circle passing through the points P(6, –6), Q(3, –7) and R (3, 3).
Points (−4, 0) and (7, 0) lie
If the point P (m, 3) lies on the line segment joining the points \[A\left( - \frac{2}{5}, 6 \right)\] and B (2, 8), find the value of m.
If \[D\left( - \frac{1}{5}, \frac{5}{2} \right), E(7, 3) \text{ and } F\left( \frac{7}{2}, \frac{7}{2} \right)\] are the mid-points of sides of \[∆ ABC\] , find the area of \[∆ ABC\] .
If P (x, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.
If (x , 2), (−3, −4) and (7, −5) are collinear, then x =
The distance of the point (4, 7) from the y-axis is
What are the coordinates of origin?
The distance of the point P(2, 3) from the x-axis is ______.
Ordinate of all points on the x-axis is ______.
If the perpendicular distance of a point P from the x-axis is 5 units and the foot of the perpendicular lies on the negative direction of x-axis, then the point P has ______.
Find the coordinates of the point which lies on x and y axes both.
