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प्रश्न
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(4, 5), (7, 6), (4, 3), (1, 2)
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उत्तर
Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B, C, and D of the given quadrilateral respectively.
∴ AB = `sqrt((4-7)^2+(5-6)^2)`
= `sqrt((-3)^2+(-1)^2)`
= `sqrt(9+1)`
= `sqrt10`
BC = `sqrt((7-4)^2+(6-3)^2)`
= `sqrt((3)^2+(3)^2)`
= `sqrt(9+9)`
= `sqrt18`
CD = `sqrt((4-1)^2+(3-2)^2)`
= `sqrt((3)^2+(1)^2)`
= `sqrt(9+1)`
= `sqrt10`
AD = `sqrt((4-1)^2+(5-2)^2)`
= `sqrt((3)^2+(3)^2)`
= `sqrt(9+9)`
= `sqrt18`
Diagonal AC = `sqrt((4-4)^2+(5-3)^2)`
= `sqrt((0)^2+(2)^2)`
= `sqrt(0+4)`
= 2
Diagonal CD = `sqrt((7-1)^2 + (6-2)^2)`
= `sqrt((6)^2+(4)^2)`
= `sqrt(36+16)`
= `sqrt52`
= `13sqrt2`
It can be observed that opposite sides of this quadrilateral are of the same length. However, the diagonals are of different lengths. Therefore, the given points are the vertices of a parallelogram.
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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.
Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.
Show that the points (1, – 1), (5, 2) and (9, 5) are collinear.
Find the distances between the following point.
P(–6, –3), Q(–1, 9)
If A and B are the points (−6, 7) and (−1, −5) respectively, then the distance
2AB is equal to
Find the distance between the following pairs of point in the coordinate plane :
(4 , 1) and (-4 , 5)
Find the distance of the following point from the origin :
(8 , 15)
Find the distance of a point (12 , 5) from another point on the line x = 0 whose ordinate is 9.
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.
x (1,2),Y (3, -4) and z (5,-6) are the vertices of a triangle . Find the circumcentre and the circumradius of the triangle.
Find the distance between the following pairs of points:
`(3/5,2) and (-(1)/(5),1(2)/(5))`
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
Point P (2, -7) is the centre of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of AB.
Use distance formula to show that the points A(-1, 2), B(2, 5) and C(-5, -2) are collinear.
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
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 ______.
If (a, b) is the mid-point of the line segment joining the points A(10, –6) and B(k, 4) and a – 2b = 18, find the value of k and the distance AB.
The distance between the points (0, 5) and (–3, 1) is ______.
Read the following passage:
|
Alia and Shagun are friends living on the same street in Patel Nagar. Shagun's house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point O, i.e., the origin, Alia's house is at A. Shagun's house is at B and library is at C. |
Based on the above information, answer the following questions.
- How far is Alia's house from Shagun's house?
- How far is the library from Shagun's house?
- Show that for Shagun, school is farther compared to Alia's house and library.
OR
Show that Alia’s house, shagun’s house and library for an isosceles right triangle.
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
