Advertisements
Advertisements
प्रश्न
Prove hat the points A (2, 3) B(−2,2) C(−1,−2), and D(3, −1) are the vertices of a square ABCD.
उत्तर
The distance d between two points (x1 , y1) and (x2 , y2) is given by the formula
`d = sqrt((x_1- x_2 )^2 + (y_1 - y_2)^2)`
In a square all the sides are equal to each other. And also the diagonals are also equal to each other.
Here the four points are A(5,6), B(1,5), C(2,1) and D(6,2).
First let us check if all the four sides are equal.
`AB = sqrt((5 - 1)^2 + (6 - 5)^2)`
`= sqrt((4)^2 + (1)^2)`
`= sqrt(16 + 1)`
`AB = sqrt(17)`
`BC = sqrt((1-2)^2 + (5 - 1)^2)`
`= sqrt((-1)^2 + (4)^2)`
`= sqrt(1 + 16)`
`BC = sqrt(17)`
`CD = sqrt((2 -6)^2 + (1- 2)^2)`
`=sqrt((-4)^2 + (-1)^2)`
`= sqrt(16+ 1)`
`CD = sqrt(17)`
`AD = sqrt((5 - 6)^2 + (6-2)^2)`
`=sqrt((-1)^2 + (4)^2)`
`= sqrt(1+16)`
`AD = sqrt(17)`
Here, we see that all the sides are equal, so it has to be a rhombus.
Now let us find out the lengths of the diagonals of this rhombus.
`AC = sqrt((5 - 2)^2 + (6-1)^2)`
`=sqrt((3)^2 + (5)^2)`
`= sqrt(9+25)`
`AC = sqrt(34)`
`BD = sqrt((1 - 6)^2 + (5 -2)^2)`
`=sqrt((-5)^2 + (3)^2)`
`= sqrt(25+9)`
`BD = sqrt(34)`
Now since the diagonals of the rhombus are also equal to each other this rhombus has to be a square.
Hence we have proved that the quadrilateral formed by the given four points is a Square.
APPEARS IN
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).
In Fig. 14.36, a right triangle BOA is given C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices O, A and B.
We have a right angled triangle,`triangle BOA` right angled at O. Co-ordinates are B (0,2b); A (2a, 0) and C (0, 0).
Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.
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.
In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (-5, -21/5)?
Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by y-axis. Also, find the coordinates of the point of division in each case.
If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.
The line segment joining the points A(3,−4) and B(1,2) is trisected at the points P(p,−2) and Q `(5/3,q)`. Find the values of p and q.
Find the coordinates of circumcentre and radius of circumcircle of ∆ABC if A(7, 1), B(3, 5) and C(2, 0) are given.
The area of the triangle formed by the points P (0, 1), Q (0, 5) and R (3, 4) is
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.
Write the distance between the points A (10 cos θ, 0) and B (0, 10 sin θ).
Write the coordinates of the point dividing line segment joining points (2, 3) and (3, 4) internally in the ratio 1 : 5.
If the points P (x, y) is equidistant from A (5, 1) and B (−1, 5), then
If the centroid of the triangle formed by the points (3, −5), (−7, 4), (10, −k) is at the point (k −1), then k =
Point (–10, 0) lies ______.
If the coordinate of point A on the number line is –1 and that of point B is 6, then find d(A, B).
If the vertices of a parallelogram PQRS taken in order are P(3, 4), Q(–2, 3) and R(–3, –2), then the coordinates of its fourth vertex S are ______.
Statement A (Assertion): If the coordinates of the mid-points of the sides AB and AC of ∆ABC are D(3, 5) and E(–3, –3) respectively, then BC = 20 units.
Statement R (Reason): The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.
