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प्रश्न
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.
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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)`
The co-ordinates of the midpoint `(x_m,y_m)` between two points `(x_1, y_1)` and `(x_2, y_2)` is given by,
`(x_m,y_m) = (((x_1 + x_2)/2)"," ((y_1+y_2)/2))`
Here, it is given that the three vertices of a triangle are A(−1,3), B(1,−1) and C(5,1).
The median of a triangle is the line joining a vertex of a triangle to the mid-point of the side opposite this vertex.
Let ‘D’ be the mid-point of the side ‘BC’.
Let us now find its co-ordinates.
`(x_D,y_D) = (((1 + 5)/2)"," ((-1+1)/2))`
`(x_D, y_D) = (3,0)`
Thus we have the co-ordinates of the point as D(3,0).
Now, let us find the length of the median ‘AD’.
`AD = sqrt((-1-3)^2 + (3 - 0)^2)`
`= sqrt((-4)^2 + (3)^2)`
`= sqrt(16 + 9)`
AD = 5
Thus the length of the median through the vertex ‘A’ of the given triangle is 5 units
संबंधित प्रश्न
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. Champa and Chameli walk into the class and after observing for a few minutes, Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees.
Using distance formula, find which of them is correct.
If Q (0, 1) is equidistant from P (5, − 3) and R (x, 6), find the values of x. Also find the distance QR and PR.
If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y
An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
Find the distance of the following points from the origin:
(i) A(5,- 12)
Find the distances between the following point.
P(–6, –3), Q(–1, 9)
Show that the ▢PQRS formed by P(2, 1), Q(–1, 3), R(–5, –3) and S(–2, –5) is a rectangle.
Find the distance of the following point from the origin :
(0 , 11)
Prove that the points (5 , 3) , (1 , 2), (2 , -2) and (6 ,-1) are the vertices of a square.
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 points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
The distance between points P(–1, 1) and Q(5, –7) is ______
If the distance between point L(x, 7) and point M(1, 15) is 10, then find the value of x
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 distance between the points O(0, 0) and P(3, 4).
Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
The distance between the points (0, 5) and (–3, 1) is ______.
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Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
- At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1, 2), B(4, 3) and C(6, 6)
- Check if the Goal keeper G(–3, 5), Sweeper H(3, 1) and Wing-back K(0, 3) fall on a same straight line.
[or]
Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ. - If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
