Advertisements
Advertisements
प्रश्न
Show that the points (0, –1), (8, 3), (6, 7) and (–2, 3) are vertices of a rectangle.
Advertisements
उत्तर
Let the points be P(0, –1), Q(8, 3), R(6, 7), S(–2, 3)
Distance between two points = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ By distance formula,
d(P, Q) = `sqrt((8 - 0)^2 + [3 - (-1)]^2`
= `sqrt((8 - 0)^2 + (3 + 1)^2`
= `sqrt(8^2 + 4^2)`
= `sqrt(64 + 16)`
= `sqrt(80)` ...(i)
d(Q, R) = `sqrt((6 - 8)^2 + (7 - 3)^2`
= `sqrt((-2)^2 + (4)^2`
= `sqrt(4 + 16)`
= `sqrt(20)` ...(ii)
d(R, S) = `sqrt([(-2) - 6]^2 + (3 - 7)^2`
= `sqrt((-8)^2 + (-4)^2`
= `sqrt(64 + 16)`
=`sqrt(80)` ...(iii)
d(P, S) = `sqrt([(-2) - 0]^2 + [3 - (-1)^2]`
= `sqrt((-2)^2+ (3+ 1)^2`
= `sqrt((-2)^2 + 4^2`
= `sqrt(4 + 16)`
= `sqrt(20)` ...(iv)
In ▢PQRS,
∴ side PQ = side RS ...[From (i) and (iii)]
side QR = side PS ...[From (ii) and (iv)]
∴ ▢PQRS is a parallelogram ...[A quadrilateral is a parallelogram, if both the pairs of its opposite sides are congruent]
d(P, R) = `sqrt((6 - 0)^2 + [7 - (-1)]^2`
= `sqrt((6 - 0)^2 + (7 + 1)^2`
= `sqrt(6^2 + 8^2)`
= `sqrt(36 + 64)`
= `sqrt(100)`
= 10 ...(iv)
d(Q, S) = `sqrt([(-2) - 8]^2 + [3 - 3]^2`
= `sqrt((-10)^2 + (0)^2`
= `sqrt(100 + 0)`
= `sqrt(100)`
= 10 ...(vi)
In parallelogram PQRS,
PR = QS ...[From (v) and (vi)]
∴ ▢PQRS is a rectangle. ...[A parallelogram is a rectangle if its diagonals are equal]
APPEARS IN
संबंधित प्रश्न
Show that the points (1, – 1), (5, 2) and (9, 5) are collinear.
If a ≠ b ≠ 0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
Find the distance between the points:
A(1, –3) and B(4, –6)
Find the distance of the following points from the origin:
A(5, –12)
Find the distance of the following points from the origin:
C(–4, –6)
Using the distance formula, show that the given points are collinear:
(1, –1), (5, 2) and (9, 5)
Find the distance between the following pair of point.
P(–5, 7), Q(–1, 3)
Find the distances between the following point.
A(a, 0), B(0, a)
Find the value of y for which the distance between the points A (3, −1) and B (11, y) is 10 units.
If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.
A line segment of length 10 units has one end at A (-4 , 3). If the ordinate of te othyer end B is 9 , find the abscissa of this end.
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.
From the given number line, find d(A, B):
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
Calculate the distance between A (7, 3) and B on the x-axis whose abscissa is 11.
The points (– 4, 0), (4, 0), (0, 3) are the vertices of a ______.
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
