Advertisements
Advertisements
Question
Show that the following points are the vertices of a rectangle.
A (2, -2), B(14,10), C(11,13) and D(-1,1)
Advertisements
Solution
The given points are A (2, -2), B(14,10), C(11,13) and D(-1,1).
`AB = sqrt((14-2)^2 +{10-(-2)}^2) = sqrt((12)^2 +(12)^2) =sqrt(144+144) = sqrt(288) =12 sqrt(2) units`
`BC = sqrt(( 11-14)^2 +(13-10)^2 ) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) = 3 sqrt(2) units`
` CD = sqrt((-1-11)^2 +(1-13)^2) = sqrt((-12)^2 +(-12)^2) = sqrt(144+144) = sqrt(288) = 12 sqrt(2) units`
`AD = sqrt((-1-2)^2 +{1-(-2)}^2) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) =3 sqrt(2) units`
`Thus AB =CD = 12 sqrt(2) "units and " BC =AD = 3 sqrt(2) units`
Also ,
`AC = sqrt((11-2)^2 +{ 13-(-2)}^2) = sqrt((9)^2 +(15)^2) = sqrt(81+225) = sqrt(306) = 3 sqrt(34) units `
` BD = sqrt((-1-14)^2 +(1-10)^2) = sqrt((-15)^2 +(-9)^2) = sqrt(81+225) = sqrt(306) =3 sqrt(34) units`
Also, diagonal AC = diagonal BD
Hence, the given points from a rectangle
APPEARS IN
RELATED QUESTIONS
Find the distance between the following pair of points:
(a, 0) and (0, b)
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.
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
Find the third vertex of a triangle, if two of its vertices are at (−3, 1) and (0, −2) and the centroid is at the origin.
Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).
The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.
Show hat A(1,2), B(4,3),C(6,6) and D(3,5) are the vertices of a parallelogram. Show that ABCD is not rectangle.
In what ratio is the line segment joining the points A(-2, -3) and B(3,7) divided by the yaxis? Also, find the coordinates of the point of division.
If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
Find the possible pairs of coordinates of the fourth vertex D of the parallelogram, if three of its vertices are A(5, 6), B(1, –2) and C(3, –2).
A point whose abscissa and ordinate are 2 and −5 respectively, lies in
The perpendicular distance of the point P (4, 3) from x-axis is
If the centroid of the triangle formed by points P (a, b), Q(b, c) and R (c, a) is at the origin, what is the value of a + b + c?
What is the distance between the points A (c, 0) and B (0, −c)?
The distance of the point (4, 7) from the y-axis is
If the points P (x, y) is equidistant from A (5, 1) and B (−1, 5), then
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
Students of a school are standing in rows and columns in their playground for a drill practice. A, B, C and D are the positions of four students as shown in figure. Is it possible to place Jaspal in the drill in such a way that he is equidistant from each of the four students A, B, C and D? If so, what should be his position?
Signs of the abscissa and ordinate of a point in the second quadrant are respectively.
A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled floor in the archaeological Museum of Seville, made using squares, triangles and hexagons.
A craftsman thought of making a floor pattern after being inspired by the above design. To ensure accuracy in his work, he made the pattern on the Cartesian plane. He used regular octagons, squares and triangles for his floor tessellation pattern
Use the above figure to answer the questions that follow:
- What is the length of the line segment joining points B and F?
- The centre ‘Z’ of the figure will be the point of intersection of the diagonals of quadrilateral WXOP. Then what are the coordinates of Z?
- What are the coordinates of the point on y-axis equidistant from A and G?
OR
What is the area of Trapezium AFGH?
