Advertisements
Advertisements
Question
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
Advertisements
Solution
The distance d between two points `(x_1,y_1)` and `(x_2, y_2)` is given by the formula
`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2`
For three points to be collinear the sum of distances between two pairs of points should be equal to the third pair of points.
The given points are A (−2, 5), B (0, 1) and C (2, −3)
Let us find the distances between the possible pairs of points.
`AB = sqrt((-2 - 0)^2 + (5 - 1)^2)`
`= sqrt((-2)^2 + (4)^2)`
`= sqrt(4 + 16)`
`AB = 2sqrt5`
`AC = sqrt((-2-2)^2 + (5 + 3)^2)`
`= sqrt((-4)^2 + (8)^2)`
`= sqrt(16 + 64)`
`AC = 4sqrt5`
`BC = sqrt((0 - 2)^2 + (1 + 3)^2)`
`= sqrt((-2)^2 + (4))`
`= sqrt(4 + 16)`
`BC = 2sqrt5`
We see that AB + BC = AC
Since sum of distances between two pairs of points equals the distance between the third pair of points the three points must be collinear.
Hence we have proved that the three given points are collinear.
APPEARS IN
RELATED QUESTIONS
Find the distance between the following pair of points:
(a, 0) and (0, b)
Which point on the y-axis is equidistant from (2, 3) and (−4, 1)?
Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).
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
If p(x , y) is point equidistant from the points A(6, -1) and B(2,3) A , show that x – y = 3
If the point A (4,3) and B ( x,5) lies on a circle with the centre o (2,3) . Find the value of x.
`"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 point P(m, 6) divides the join of A(-4, 3) and B(2, 8) Also, find the value of m.
In what ratio is the line segment joining A(2, -3) and B(5, 6) divide by the x-axis? Also, find the coordinates of the pint of division.
In what ratio does the line x - y - 2 = 0 divide the line segment joining the points A (3, 1) and B (8, 9)?
The abscissa of any point on y-axis is
what is the value of \[\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}\] .
If the centroid of the triangle formed by (7, x) (y, −6) and (9, 10) is at (6, 3), then (x, y) =
The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are
The line 3x + y – 9 = 0 divides the line joining the points (1, 3) and (2, 7) internally in the ratio ______.
The coordinates of a point whose ordinate is `-1/2` and abscissa is 1 are `-1/2, 1`.
Find the coordinates of the point whose ordinate is – 4 and which lies on y-axis.
Seg AB is parallel to X-axis and coordinates of the point A are (1, 3), then the coordinates of the point B can be ______.
Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x-axis is 1:2.
Reason (R): as formula for the internal division is `((mx_2 + nx_1)/(m + n) , (my_2 + ny_1)/(m + n))`
Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So, he started to sketch his own rocket designs on the graph sheet. One such design is given below :
Based on the above, answer the following questions:
i. Find the mid-point of the segment joining F and G. (1)
ii. a. What is the distance between the points A and C? (2)
OR
b. Find the coordinates of the points which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (2)
iii. What are the coordinates of the point D? (1)
