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प्रश्न
Find the distance between the following pairs of points:
(2, 3), (4, 1)
Find the distance between the following pairs of points:
A (2, 3), B (4, 1)
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उत्तर १
l = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
= `sqrt((4 - 2)^2 + (1 - 3)^2)`
= `sqrt(2^2 + (- 2)^2)`
= `sqrt(4 + 4)`
= `sqrt(8)`
= `sqrt(4 × 2)`
= `2sqrt(2)` units
उत्तर २
A (2, 3), B (4, 1)
Suppose the coordinates of point A are (x1, y1) and those of point B are (x2, y2).
x1 = 2, y1 = 3, x2 = 4, y2 = 1
According to distance formula,
d(A,B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
d(A,B) = `sqrt((4 - 2)^2 + (1 - 3)^2)`
d(A,B) = `sqrt(2^2 + (- 2)^2)`
d(A,B) = `sqrt(4 + 4)`
d(A,B) = `sqrt(8)`
d(A,B) = `sqrt(4 × 2)`
d(A,B) = `2sqrt(2)`
The distance between points A and B is `2sqrt(2)` units.
संबंधित प्रश्न
If P (2, – 1), Q(3, 4), R(–2, 3) and S(–3, –2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus
Find the distance between the following pairs of points:
(−5, 7), (−1, 3)
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.
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
Prove that the points A(1, 7), B(4, 2), C(−1, −1), and D(−4, 4) are the vertices of a square.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
Find the distance between the points:
P(a + b, a – b) and Q(a – b, a + b)
Find all possible values of y for which the distance between the points A(2, –3) and B(10, y) is 10 units.
If the point A(x, 2) is equidistant from the points B(8, –2) and C(2, –2), find the value of x. Also, find the length of AB.
Find the distances between the following point.
A(a, 0), B(0, a)
Find the distance between the following point :
(p+q,p-q) and (p-q, p-q)
Find the value of m if the distance between the points (m , -4) and (3 , 2) is 3`sqrt 5` units.
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
A line segment of length 10 units has one end at A (-4 , 3). If the ordinate of te othyer end B is 9 , find the abscissa of this end.
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.
Prove that the points (1 , 1) , (-1 , -1) and (`- sqrt 3 , sqrt 3`) are the vertices of an equilateral triangle.
Prove that the points (4 , 6) , (- 1 , 5) , (- 2, 0) and (3 , 1) are the vertices of a rhombus.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.
Find the distance between the points (a, b) and (−a, −b).
Find the distance between the following pairs of points:
`(3/5,2) and (-(1)/(5),1(2)/(5))`
The distance between the points (3, 1) and (0, x) is 5. Find x.
Show that (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus.
Points A (-3, -2), B (-6, a), C (-3, -4) and D (0, -1) are the vertices of quadrilateral ABCD; find a if 'a' is negative and AB = CD.
Use distance formula to show that the points A(-1, 2), B(2, 5) and C(-5, -2) are collinear.
Find distance between point A(–1, 1) and point B(5, –7):
Solution: Suppose A(x1, y1) and B(x2, y2)
x1 = –1, y1 = 1 and x2 = 5, y2 = –7
Using distance formula,
d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ d(A, B) = `sqrt(square +[(-7) + square]^2`
∴ d(A, B) = `sqrt(square)`
∴ d(A, B) = `square`
Show that the point (0, 9) is equidistant from the points (–4, 1) and (4, 1).
Using distance formula decide whether the points (4, 3), (5, 1) and (1, 9) are collinear or not.
The distance between the points (0, 5) and (–5, 0) is ______.
The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) is ______.
If the distance between the points (4, P) and (1, 0) is 5, then the value of p is ______.
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:
The coordinates of the centroid of ΔEHJ are ______.
Point P(0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A(–1, 1) and B(3, 3).
Points A(4, 3), B(6, 4), C(5, –6) and D(–3, 5) are the vertices of a parallelogram.
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
The distance of the point (5, 0) from the origin is ______.
