Advertisements
Advertisements
Question
Prove that the points A(-4,-1), B(-2, 4), C(4, 0) and D(2, 3) are the vertices of a rectangle.
Advertisements
Solution 1
Let A (-4,-1); B (-2,-4); C (4, 0) and D (2, 3) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rectangle.
So we should find the lengths of opposite sides of quadrilateral ABCD.
`AB = sqrt((-2+4)^2) + (-4 + 1)^2)`
`= sqrt(4 + 9)`
`= sqrt13`
`CD = sqrt((4 - 2)^2 + (0 - 3)^2)`
`= sqrt(4 +9)`
`= sqrt13`
Opposite sides are equal. So now we will check the lengths of the diagonals.
`AC = sqrt((4 + 4)^2 + (0 + 1)^2)`
`= sqrt(64 + 1)`
`= sqrt(65)`
`BD = sqrt((2 + 2)^2 + (3 + 4)^2)`
`= sqrt(16 + 49)`
`= sqrt65`
Opposite sides are equal as well as the diagonals are equal. Hence ABCD is a rectangle.
Solution 2
The given points are A (-4,-1); B (-2,-4); C (4, 0) and D (2, 3) .
`AB = sqrt({-2-(-4)}^2 + { -4-(-1)}^2) = sqrt ((2)^2+(-3)^2) = sqrt(4+9) = sqrt(13) ` units
` BC = sqrt({ 4-(-2)}^2+{0-(-4)}^2) = sqrt((6)^2 +(4)^2) = sqrt(36+16) = sqrt(52) = 2 sqrt(13) units`
`CD = sqrt((2-4)^2 +(3-0)^2) = sqrt((-2)^2 +(3)^2) = sqrt(4+9) = sqrt(13) units`
`AD = sqrt({2-(-4)}^2 + {3-(-1)}^2) = sqrt((6)^2 +(4)^2) = sqrt(36+16) = sqrt(52) = 2 sqrt(13) units`
`Thus , AB = CD = sqrt(13) units and BC = AD = 2 sqrt(13) units`
Also , `AC = sqrt({4-(-4)}^2+{0-(-1)}^2) = sqrt ((8)^2+(1)^2 ) = sqrt(64+1) = sqrt(65) units`
`BD = sqrt({2-(-2)}^2 +{3-(-4)}^2) = sqrt((4)^2 +(7)^2) = sqrt(16+49) = sqrt(65) units`
Also, diagonal AC = diagonal BD
Hence, the given points form a rectanglr
RELATED QUESTIONS
Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.
(Street Plan): A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction.
All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction. Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:
- how many cross - streets can be referred to as (4, 3).
- how many cross - streets can be referred to as (3, 4).
Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).
Find a point on y-axis which is equidistant from the points (5, -2) and (-3, 2).
In what ratio is the line segment joining the points (-2,-3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.
Points P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R.
Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.
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).
Find the value of a so that the point (3, a) lies on the line represented by 2x − 3y + 5 = 0
The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ − b cos θ) is
The line segment joining points (−3, −4), and (1, −2) is divided by y-axis in the ratio.
The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is
If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a3 + b3 + c3 =
In Fig. 14.46, the area of ΔABC (in square units) is
The line segment joining the points A(2, 1) and B (5, - 8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x - y + k= 0 find the value of k.
The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that AR=34AB.">AR = `3/4`AB. Find the coordinates of R.
Find the coordinates of the point of intersection of the graph of the equation x = 2 and y = – 3
Point (–10, 0) lies ______.
The point whose ordinate is 4 and which lies on y-axis is ______.
Find the coordinates of the point which lies on x and y axes both.
