Advertisements
Advertisements
प्रश्न
Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.
Advertisements
उत्तर
Let A(x1, y1) = A(4, 3), B(x2, y2) = B(5, 1), C(x3, y3) = C(1, 9)
∴ d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt((5 -4)^2 + (1 - 3)^2`
= `sqrt(1^2 + (-2)^2`
= `sqrt(1+ 4)`
= `sqrt(5)` ......(i)
∴ d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2`
= `sqrt((1 - 5)^2 + (9 - 1)^2`
= `sqrt((-4)^2 + 8^2`
= `sqrt(16 + 64)`
=`sqrt(80)`
= `4sqrt(5)` ......(ii)
∴ d(A, C) = `sqrt((x_3 -x_1)^2 + (y_3 - y_2)^2`
= `sqrt((1 - 4)^2 + (9 - 3)^2`
= `sqrt((-3)^2 + 6^2`
= `sqrt(9 + 36)`
= `sqrt(45)`
= `3sqrt(5)` .....(iii)
`sqrt(5) + 3sqrt(5) = 4sqrt(5)`
∴ d(A, B) + d(A, C) = d(B, C) ......[From (i), (ii) and (iii)]
∴ Points A, B, C are collinear.
APPEARS IN
संबंधित प्रश्न
If A(5, 2), B(2, −2) and C(−2, t) are the vertices of a right angled triangle with ∠B = 90°, then find the value of t.
Determine if the points (1, 5), (2, 3) and (−2, −11) are collinear.
ABC is a triangle and G(4, 3) is the centroid of the triangle. If A = (1, 3), B = (4, b) and C = (a, 1), find ‘a’ and ‘b’. Find the length of side BC.
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Find the distance of the following points from the origin:
(ii) B(-5,5)
Find the distance between the following pair of points.
R(0, -3), S(0, `5/2`)
Determine whether the points are collinear.
P(–2, 3), Q(1, 2), R(4, 1)
Find the distances between the following point.
A(a, 0), B(0, a)
AB and AC are the two chords of a circle whose radius is r. If p and q are
the distance of chord AB and CD, from the centre respectively and if
AB = 2AC then proove that 4q2 = p2 + 3r2.
Find the value of m if the distance between the points (m , -4) and (3 , 2) is 3`sqrt 5` units.
Prove that the points (1 ,1),(-4 , 4) and (4 , 6) are the certices of an isosceles triangle.
Prove that the points (1 , 1) , (-1 , -1) and (`- sqrt 3 , sqrt 3`) are the vertices of an equilateral triangle.
Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram.
From the given number line, find d(A, B):
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Seg OA is the radius of a circle with centre O. The coordinates of point A is (0, 2) then decide whether the point B(1, 2) is on the circle?
The coordinates of the point which is equidistant from the three vertices of the ΔAOB as shown in the figure is ______.
Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
Find the points on the x-axis which are at a distance of `2sqrt(5)` from the point (7, – 4). How many such points are there?
The point P(–2, 4) lies on a circle of radius 6 and centre C(3, 5).
