Advertisements
Advertisements
प्रश्न
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
Advertisements
उत्तर
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
संबंधित प्रश्न
(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).
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(-3, 5) B(3, 1), C (0, 3), D(-1, -4)
If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.
In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?
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.
If the point C ( - 2,3) is equidistant form the points A (3, -1) and Bx (x ,8) , find the value of x. Also, find the distance between BC
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.
Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and D(2,6) are equal and bisect each other
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).
The area of the triangle formed by the points A(2,0) B(6,0) and C(4,6) is
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).
If P (2, 6) is the mid-point of the line segment joining A(6, 5) and B(4, y), find y.
If P (x, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.
If x is a positive integer such that the distance between points P (x, 2) and Q (3, −6) is 10 units, then x =
If the centroid of the triangle formed by the points (3, −5), (−7, 4), (10, −k) is at the point (k −1), then k =
Write the equations of the x-axis and y-axis.
What are the coordinates of origin?
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?
Distance of the point (6, 5) from the y-axis is ______.
