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?
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
Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.
Determine whether the points are collinear.
A(1, −3), B(2, −5), C(−4, 7)
Find the distance of a point (13 , -9) from another point on the line y = 0 whose abscissa is 1.
Find the coordinate of O , the centre of a circle passing through P (3 , 0), Q (2 , `sqrt 5`) and R (`-2 sqrt 2` , -1). Also find its radius.
x (1,2),Y (3, -4) and z (5,-6) are the vertices of a triangle . Find the circumcentre and the circumradius of the triangle.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
A point A is at a distance of `sqrt(10)` unit from the point (4, 3). Find the co-ordinates of point A, if its ordinate is twice its abscissa.
Points A (-3, -2), B (-6, a), C (-3, -4) and D (0, -1) are the vertices of quadrilateral ABCD; find a if 'a' is negative and AB = CD.
Show that A(1, 2), (1, 6), C(1 + 2 `sqrt(3)`, 4) are vertices of a equilateral triangle
Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.
