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प्रश्न
If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.
पर्याय
AP = \[\frac{1}{3}\text{AB}\]
AP = PB
PB = \[\frac{1}{3}\text{AB}\]
- AP = \[\frac{1}{2}\text{AB}\]
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उत्तर
If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then `underlinebb(AP = 1/2 AB)`.
Explanation:
Given that, the point P(2, 1) lies on the line segment joining the points A(4, 2) and B(8, 4), which shows in the figure below:
Now, distance between A(4, 2) and P(2, 1),
AP = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
AP = `sqrt((2 - 4)^2 + (1 -2)^2`
= `sqrt((-2)^2 + (-1)^2`
= `sqrt(4 + 1)`
= `sqrt(5)`
Distance between A(4, 2) and B(8, 4),
AB = `sqrt((8 - 4)^2 + (4 - 2)^2`
= `sqrt((4)^2 + (2)^2`
= `sqrt(16 + 4)`
= `sqrt(20)`
= `2sqrt(5)`
Distance between B(8, 4) and P(2, 1),
BP = `sqrt((8 - 2)^2 + (4 - 1)^2`
= `sqrt(6^2 + 3^2`
= `sqrt(36 + 9)`
= `sqrt(45)`
= `3sqrt(5)`
∴ AB = `2sqrt(5)`
= 2AP
⇒ AP = `"AB"/2`
Hence, required condition is AP = `"AB"/2`
संबंधित प्रश्न
Find the distance between two points
(i) P(–6, 7) and Q(–1, –5)
(ii) R(a + b, a – b) and S(a – b, –a – b)
(iii) `A(at_1^2,2at_1)" and " B(at_2^2,2at_2)`
Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.
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.
Find the distance between the following pair of points:
(-6, 7) and (-1, -5)
Find the distance between the following pair of points:
(a+b, b+c) and (a-b, c-b)
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.
Find the distance between the following pair of point.
P(–5, 7), Q(–1, 3)
Find the distance between the following pair of points.
L(5, –8), M(–7, –3)
Prove that the following set of point is collinear :
(4, -5),(1 , 1),(-2 , 7)
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
A(-2, -3), B(-1, 0) and C(7, -6) are the vertices of a triangle. Find the circumcentre and the circumradius of the triangle.
A point P (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.
Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.
Given A = (x + 2, -2) and B (11, 6). Find x if AB = 17.
The distances of point P (x, y) from the points A (1, - 3) and B (- 2, 2) are in the ratio 2: 3.
Show that: 5x2 + 5y2 - 34x + 70y + 58 = 0.
Show that P(–2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle.
Show that A(1, 2), (1, 6), C(1 + 2`sqrt(3)`, 4) are vertices of an equilateral triangle.
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 x axis equidistant from I and E is ______.
Name the type of triangle formed by the points A(–5, 6), B(–4, –2) and C(7, 5).
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
