Advertisements
Advertisements
Question
Show that A (−3, 2), B (−5, −5), C (2,−3), and D (4, 4) are the vertices of a rhombus.
Advertisements
Solution
Let A (−3, 2); B (−5,−5); C (2,−3) and D (4, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rhombus.
So we should find the lengths of sides of quadrilateral ABCD.
`AB = sqrt((-5 + 3 )^2 + (-5-2)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
`BC = sqrt((2 + 5 )^2 + (-3+ 5)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
`CD = sqrt(( 4 - 2 )^2 + (4 +3)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
`AD= sqrt((4 + 3 )^2 + (4-2)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
All the sides of quadrilateral are equal. Hence ABCD is a rhombus.
APPEARS IN
RELATED QUESTIONS
If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides
Which point on the y-axis is equidistant from (2, 3) and (−4, 1)?
Find a point on y-axis which is equidistant from the points (5, -2) and (-3, 2).
Find the points of trisection of the line segment joining the points:
(3, -2) and (-3, -4)
Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet
Prove that the points A(-4,-1), B(-2, 4), C(4, 0) and D(2, 3) are the vertices of a rectangle.
Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that `(PA)/( PQ)=2/5` . If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k.
If (2, p) is the midpoint of the line segment joining the points A(6, -5) and B(-2,11) find the value of p.
`"Find the ratio in which the poin "p (3/4 , 5/12) " divides the line segment joining the points "A (1/2,3/2) and B (2,-5).`
Find the ratio in which the pint (-3, k) divide the join of A(-5, -4) and B(-2, 3),Also, find the value of k.
If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
If the point C(k,4) divides the join of A(2,6) and B(5,1) in the ratio 2:3 then find the value of k.
If (a,b) is the mid-point of the line segment joining the points A (10, - 6) , B (k,4) and a - 2b = 18 , find the value of k and the distance AB.
Write the coordinates of the point dividing line segment joining points (2, 3) and (3, 4) internally in the ratio 1 : 5.
If A (2, 2), B (−4, −4) and C (5, −8) are the vertices of a triangle, than the length of the median through vertex C is
What is the form of co-ordinates of a point on the X-axis?
Find the point on the y-axis which is equidistant from the points (S, - 2) and (- 3, 2).
Write the equations of the x-axis and y-axis.
What are the coordinates of origin?
Find the coordinates of the point whose abscissa is 5 and which lies on x-axis.
