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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 A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
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.
If the point P(2, 2) is equidistant from the points A(–2, k) and B(–2k, –3), find k. Also, find the length of AP.
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)
Find the distance of a point P(x, y) from the origin.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
Find the distance between the following pairs of point.
W `((- 7)/2 , 4)`, X (11, 4)
Prove that the following set of point is collinear :
(5 , 1),(3 , 2),(1 , 3)
The centre of a circle passing through P(8, 5) is (x+l , x-4). Find the coordinates of the centre if the diameter of the circle is 20 units.
ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.
What point on the x-axis is equidistant from the points (7, 6) and (-3, 4)?
Find distance between points O(0, 0) and B(–5, 12).
Find distance CD where C(–3a, a), D(a, –2a).
The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) 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 ______.
What is the distance of the point (– 5, 4) from the origin?
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In a GPS, The lines that run east-west are known as lines of latitude, and the lines running north-south are known as lines of longitude. The latitude and the longitude of a place are its coordinates and the distance formula is used to find the distance between two places. The distance between two parallel lines is approximately 150 km. A family from Uttar Pradesh planned a round trip from Lucknow (L) to Puri (P) via Bhuj (B) and Nashik (N) as shown in the given figure below.
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Based on the above information answer the following questions using the coordinate geometry.
- Find the distance between Lucknow (L) to Bhuj (B).
- If Kota (K), internally divide the line segment joining Lucknow (L) to Bhuj (B) into 3 : 2 then find the coordinate of Kota (K).
- Name the type of triangle formed by the places Lucknow (L), Nashik (N) and Puri (P)
[OR]
Find a place (point) on the longitude (y-axis) which is equidistant from the points Lucknow (L) and Puri (P).
Find the distance between the points O(0, 0) and P(3, 4).
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
