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प्रश्न
Show that the points A(1, 2), B(1, 6), C(1 + 2`sqrt3`, 4) are vertices of an equilateral triangle.
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उत्तर
The given points are A(1, 2), B(1, 6), C(1 + 2`sqrt3`, 4).
`"Distance between" = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
By distance formula,
AB = `sqrt((1 - 1)^2 + (6 - 2)^2)`
∴ AB = `sqrt((0)^2 + (4)^2)`
∴ AB = `sqrt(0 + 16)`
∴ AB = `sqrt(16)`
∴ AB = 4 ...(1)
BC = `sqrt((1 + 2sqrt3 - 1)^2 + (4 - 6)^2)`
∴ BC = `sqrt((2sqrt3)^2 + (-2)^2)`
∴ BC = `sqrt(12 + 4)`
∴ BC = `sqrt(16)`
∴ BC = 4 ...(2)
AC = `sqrt((1 + 2sqrt3 - 1)^2 + (4 - 2)^2)`
∴ AC = `sqrt((2sqrt3)^2 + (2)^2)`
∴ AC = `sqrt(12 + 4)`
∴ AC = `sqrt(16)`
∴ AC = 4 ...(3)
From (1), (2) and (3)
∴ AB = BC = AC = 4
Since, all the sides of equilateral triangle are congruent.
∴ ΔABC is an equilateral triangle.
The points A, B and C are the vertices of an equilateral triangle.
संबंधित प्रश्न
If two vertices of an equilateral triangle be (0, 0), (3, √3 ), find the third vertex
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Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)
If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y
If the points (2, 1) and (1, -2) are equidistant from the point (x, y), show that x + 3y = 0.
Find the values of x, y if the distances of the point (x, y) from (-3, 0) as well as from (3, 0) are 4.
Determine whether the point is collinear.
R(0, 3), D(2, 1), S(3, –1)
Find the distance of the following point from the origin :
(5 , 12)
Find the coordinate of O , the centre of a circle passing through P (3 , 0), Q (2 , `sqrt 5`) and R (`-2 sqrt 2` , -1). Also find its radius.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Find the distance between the following pairs of points:
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A point A is at a distance of `sqrt(10)` unit from the point (4, 3). Find the co-ordinates of point A, if its ordinate is twice its abscissa.
The length of line PQ is 10 units and the co-ordinates of P are (2, -3); calculate the co-ordinates of point Q, if its abscissa is 10.
Find distance between points O(0, 0) and B(– 5, 12)
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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:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
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:
The point on y axis equidistant from B and C is ______.
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
