Advertisements
Advertisements
Question
What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?
Advertisements
Solution
The points are A(2, –2), B(7, 3), C(11, –1) and D(6, –6)
Using distance formula,
d = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
AB = `sqrt((7 - 2)^2 + (3 + 2)^2`
= `sqrt((5)^2 + (5)^2`
= `sqrt(25 + 25)`
= `sqrt(50)`
= 5`sqrt(2)`
BC = `sqrt((11 - 7)^2 + (-1 - 3)^2`
= `sqrt((4)^2 + (-4)^2`
= `sqrt(16 + 16)`
= `sqrt(32)`
= `4sqrt(2)`
CD = `sqrt((6 - 11)^2 + (-6 + 1)^2`
= `sqrt((-5)^2 + (-5)^2`
= `sqrt(25 + 25)`
= `sqrt(50)`
= `5sqrt(2)`
DA = `sqrt((2 - 6)^2 + (-2 + 6)^2`
= `sqrt((-4)^2 + (4)^2`
= `sqrt(16 + 16)`
= `sqrt(32)`
= `4sqrt(2)`
Finding diagonals AC and BD, we get,
AC = `sqrt((11 - 2)^2 + (-1 + 2)^2`
= `sqrt((9)^2 + (1)^2`
= `sqrt(81 + 1)`
= `sqrt(82)`
And BD = `sqrt((6 - 7)^2 + (-6 - 3)^2`
= `sqrt((-1)^2 + (-9)^2`
= `sqrt(1 + 81)`
= `sqrt(82)`
The quadrilateral formed is rectangle.
APPEARS IN
RELATED QUESTIONS
If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.
If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y
A(–8, 0), B(0, 16) and C(0, 0) are the vertices of a triangle ABC. Point P lies on AB and Q lies on AC such that AP : PB = 3 : 5 and AQ : QC = 3 : 5. Show that : PQ = `3/8` BC.
Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.
Determine whether the point is collinear.
R(0, 3), D(2, 1), S(3, –1)
If A and B are the points (−6, 7) and (−1, −5) respectively, then the distance
2AB is equal to
Find the distance between the following pair of point in the coordinate plane :
(5 , -2) and (1 , 5)
Find the distance of the following point from the origin :
(5 , 12)
Find the distance of the following point from the origin :
(13 , 0)
Prove that the following set of point is collinear :
(5 , 1),(3 , 2),(1 , 3)
Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
Find the distance between the following pair of points:
`(sqrt(3)+1,1)` and `(0, sqrt(3))`
Show that the points P (0, 5), Q (5, 10) and R (6, 3) are the vertices of an isosceles triangle.
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
If the length of the segment joining point L(x, 7) and point M(1, 15) is 10 cm, then the value of x is ______.
The distance between the point P(1, 4) and Q(4, 0) is ______.
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
The point P(–2, 4) lies on a circle of radius 6 and centre C(3, 5).
