Advertisements
Advertisements
प्रश्न
Show that A(1, 2), (1, 6), C(1 + 2`sqrt(3)`, 4) are vertices of an equilateral triangle.
Advertisements
उत्तर
Distance between two points = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
By distance formula,
AB = `sqrt((1- 1)^2 + (6 - 2)^2`
= `sqrt(0^2 + 4^2)`
= `sqrt(4^2)`
= 4 ...(i)
BC = `sqrt((1 + 2sqrt(3) - 1)^2 + (4 - 6)^2`
= `sqrt((2sqrt(3))^2 + (-2)^2`
= `sqrt(12 + 4)`
= `sqrt(16)`
= 4 ...(ii)
AC = `sqrt((1 + 2sqrt(3) - 1)^2 + (4 -2)^2`
= `sqrt((2sqrt(3))^2 + 2^2`
= `sqrt(12 + 4)`
= `sqrt(16)`
= 4 ...(iii)
∴ AB = BC = AC ...[From (i), (ii) and (iii)]
∴ ∆ABC is an equilateral triangle.
∴ Points A, B and C are the vertices of an equilateral triangle.
APPEARS IN
संबंधित प्रश्न
Find the value of x, if the distance between the points (x, – 1) and (3, 2) is 5.
Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
If the point P(x, y) is equidistant from the points A (5, 1) and B (1, 5), prove that x = y.
If A (–1, 3), B (1, –1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.
Find the distance between the points:
A(7, –4) and B(–5, 1)
Find the distance of the following points from the origin:
B(–5, 5)
For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?
Find the distance between the following pairs of point.
W `((- 7)/2 , 4)`, X (11, 4)
Find the distance between the following pair of point in the coordinate plane.
(1 , 3) and (3 , 9)
Find the distance of a point (13 , -9) from another point on the line y = 0 whose abscissa is 1.
Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
Prove that the points (5 , 3) , (1 , 2), (2 , -2) and (6 ,-1) are the vertices of a square.
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
Given A = (x + 2, -2) and B (11, 6). Find x if AB = 17.
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Show that each of the triangles whose vertices are given below are isosceles :
(i) (8, 2), (5,-3) and (0,0)
(ii) (0,6), (-5, 3) and (3,1).
The distance between the point P(1, 4) and Q(4, 0) 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 ______.
