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प्रश्न
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(4, 5) B(7, 6), C (4, 3), D(1, 2)
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उत्तर
A (4, 5), B (7,6), C(4,3), D(1,2)
Let A, B, C and D be the four vertices of the quadrilateral ABCD.
We know the distance between two points `P(x_1,y_1)` and `Q(x_2, y_2)is given by distance formula:
`PQ = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
Hence
`=> AB = sqrt((7 - 4)^2 + (6 - 5)^2)`
`=> AB = sqrt((-3)^2 + (-3)^2)`
`=> AB = sqrt(9 + 1)`
`=> AB = sqrt(10)`
Similarly,
`=> BC = sqrt((4 - 7)^2 + (3 - 6)^2)`
`=> BC = sqrt((-3)^2 + (1)^2)`
`=> BC= sqrt(9 + 9)`
`=> BC = sqrt18`
Similarly
`=> CD = sqrt((1 - 4)^2 + (2 - 3)^2)`
`=> CD = sqrt((-3)^2 + (-1)^2)`
`=> CD = sqrt(9 + 1)`
`=> CD = sqrt(9 + 1)`
`=> CD = sqrt10`
Also
`=> DA = sqrt((1 - 4)^2 + (2 -5)^2)`
`=> DA = sqrt((-3)^2 + (-3)^2)`
`=> DA = sqrt(9 + 9)`
`=> DA = sqrt18`
Hence from above we see that
AB = CD and BC = DA
Hence from above we see that
AB = CD and BC = DA
Here opposite sides of the quadrilateral is equal. Hence it is a parallelogram
संबंधित प्रश्न
(Street Plan): A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction.
All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction. Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:
- how many cross - streets can be referred to as (4, 3).
- how many cross - streets can be referred to as (3, 4).
On which axis do the following points lie?
Q(0, -2)
Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.
Which point on the y-axis is equidistant from (2, 3) and (−4, 1)?
Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).
Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.
Find the points of trisection of the line segment joining the points:
(3, -2) and (-3, -4)
In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.
Show that the following points are the vertices of a square:
A (6,2), B(2,1), C(1,5) and D(5,6)
The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.
The abscissa and ordinate of the origin are
The measure of the angle between the coordinate axes is
Find the ratio in which the line segment joining the points A(3, −3) and B(−2, 7) is divided by the x-axis. Also, find the coordinates of the point of division.
Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.
The coordinates of the point on X-axis which are equidistant from the points (−3, 4) and (2, 5) are
The area of the triangle formed by (a, b + c), (b, c + a) and (c, a + b)
The line segment joining the points A(2, 1) and B (5, - 8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x - y + k= 0 find the value of k.
If segment AB is parallel Y-axis and coordinates of A are (1, 3), then the coordinates of B are ______
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 ______.
Assertion (A): Mid-point of a line segment divides the line segment in the ratio 1 : 1
Reason (R): The ratio in which the point (−3, k) divides the line segment joining the points (− 5, 4) and (− 2, 3) is 1 : 2.
