Advertisements
Advertisements
Question
Show that the points A(2,1), B(5,2), C(6,4) and D(3,3) are the angular points of a parallelogram. Is this figure a rectangle?
Advertisements
Solution
The given points are s A(2,1), B(5,2), C(6,4) and D(3,3)
`AB = sqrt((5-2)^2 +(2-1)^2 ) = sqrt((3)^2 +(1)^2 ) = sqrt(9+1) = sqrt(10) ` units
`BC = sqrt((6-5)^2 +(4-2)^2 )= sqrt((1)^2 +(2)^3) = sqrt(1+4) = sqrt(5) `units
`CD = sqrt((3-6)^2 +(3-4)^2) = sqrt((-3)^2 +(-1)^2) = sqrt(9+1) = sqrt(10) `units
`AD = sqrt((3-2)^2+(3-1)^2) = sqrt((1)^2 +(2)^2) = sqrt(1+4) = sqrt(5) ` units
Thus, AB = CD = `sqrt(10) "units and " BC= AD = sqrt(5) ` units
So, quadrilateral ABCD is a parallelogram
`Also , AC = sqrt((6-2)^2 +(4-1)^2) = sqrt((4)^2 +(3)^2 )= sqrt(16+9) = sqrt(25) = 5 ` units
`BD = sqrt((3-5) ^2 +(3-2)^2 ) = sqrt((-2)^2 +(1)^2) = sqrt(4+1) = sqrt(5) units `
But diagonal AC is not equal to diagonal BD. Hence, the given points do not form a rectangle.
APPEARS IN
RELATED QUESTIONS
Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.
The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.
Find the centre of the circle passing through (5, -8), (2, -9) and (2, 1).
Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8) respectively.
The line joining the points (2, 1) and (5, −8) is trisected at the points P and Q. If point P lies on the line 2x − y + k = 0. Find the value of k.
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.
Find the area of quadrilateral ABCD whose vertices are A(-3, -1), B(-2,-4) C(4,-1) and D(3,4)
Find the area of quadrilateral ABCD whose vertices are A(-5, 7), B(-4, -5) C(-1,-6) and D(4,5)
Find the point on x-axis which is equidistant from points A(-1,0) and B(5,0)
Find the centroid of ΔABC whose vertices are A(2,2) , B (-4,-4) and C (5,-8).
Show that A(-4, -7), B(-1, 2), C(8, 5) and D(5, -4) are the vertices of a
rhombus ABCD.
Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), B(2 + \[\sqrt{3}\] , 5) and C(2, 6).
what is the value of \[\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}\] .
Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).
If three points (0, 0), \[\left( 3, \sqrt{3} \right)\] and (3, λ) form an equilateral triangle, then λ =
The line segment joining points (−3, −4), and (1, −2) is divided by y-axis in the ratio.
If the points(x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =
The ratio in which the line segment joining points A (a1, b1) and B (a2, b2) is divided by y-axis is
If point P is midpoint of segment joining point A(–4, 2) and point B(6, 2), then the coordinates of P are ______.
The distance of the point P(2, 3) from the x-axis is ______.
