Advertisements
Advertisements
Question
Show that the points A(3,0), B(4,5), C(-1,4) and D(-2,-1) are the vertices of a rhombus. Find its area.
Solution
The given points are A(3,0), B(4,5), C(-1,4) and D(-2,-1)
`AB = sqrt((3-4)^2 + (0-5)^2 ) = sqrt((-1)^2 +(-5)^2)`
`= sqrt(1+25) = sqrt(26)`
`BC = sqrt((4+1)^2 +(5-4)^2) = sqrt((5)^2 +(1)^2)`
`= sqrt(25+1) = sqrt(26)`
`CD = sqrt((-1+2)^2 +(4+1)^2) = sqrt((1)^2 +(5)^2)`
`= sqrt(1+25) = sqrt(26)`
`AD = sqrt((3+2)^2 +(0+1)^2) = sqrt((5)^2 +(1)^2)`
`= sqrt(25+1) = sqrt(26)`
`AC = sqrt((3+1)^2 + (0-4)^2) = sqrt((4)^2+(-4)^2)`
`= sqrt(16+16) =4sqrt(2)`
`BD = sqrt((4+2)^2 +(5+1)^2 ) = sqrt((6)^2+(6)^2)`
`= sqrt((36+36)) = 6 sqrt(2) `
`∵ AB = BC =CD =AD = 6 sqrt(2) and AC ≠ BD `
Therefore, the given points are the vertices of a rhombus
Area (Δ ABCD ) =`1/2 xx AC xxBD`
`= 1/2 xx 4 sqrt(2) xx 6 sqrt(2 ) = 24 ` sq. units
Hence, the area of the rhombus is 24 sq. units.
APPEARS IN
RELATED QUESTIONS
A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.
On which axis do the following points lie?
Q(0, -2)
Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).
Show that the following points are the vertices of a square:
(i) A (3,2), B(0,5), C(-3,2) and D(0,-1)
Points P, Q, and R in that order are dividing line segment joining A (1,6) and B(5, -2) in four equal parts. Find the coordinates of P, Q and R.
Find the coordinates of the midpoints of the line segment joining
A(3,0) and B(-5, 4)
Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.
Find the value of k, if the points A(7, −2), B (5, 1) and C (3, 2k) are collinear.
Find the value of a for which the area of the triangle formed by the points A(a, 2a), B(−2, 6) and C(3, 1) is 10 square units.
If the mid-point of the segment joining A (x, y + 1) and B (x + 1, y + 2) is C \[\left( \frac{3}{2}, \frac{5}{2} \right)\] , find x, y.
Write the formula for the area of the triangle having its vertices at (x1, y1), (x2, y2) and (x3, y3).
What is the distance between the points A (c, 0) and B (0, −c)?
If P (x, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.
The perimeter of the triangle formed by the points (0, 0), (0, 1) and (0, 1) is
If Points (1, 2) (−5, 6) and (a, −2) are collinear, then a =
The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (−3, 4) are
If the sum of X-coordinates of the vertices of a triangle is 12 and the sum of Y-coordinates is 9, then the coordinates of centroid are ______
If y-coordinate of a point is zero, then this point always lies ______.
Find the coordinates of the point which lies on x and y axes both.
Assertion (A): Mid-point of a line segment divides the line segment in the ratio 1 : 1
Reason (R): The ratio in which the point (−3, k) divides the line segment joining the points (− 5, 4) and (− 2, 3) is 1 : 2.
