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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 the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.
Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
Using the distance formula, show that the given points are collinear:
(-1, -1), (2, 3) and (8, 11)
Using the distance formula, show that the given points are collinear:
(-2, 5), (0,1) and (2, -3)
Find the distances between the following point.
P(–6, –3), Q(–1, 9)
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.
Find the distance of the following point from the origin :
(5 , 12)
Find the value of m if the distance between the points (m , -4) and (3 , 2) is 3`sqrt 5` units.
Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Prove that the points (4 , 6) , (- 1 , 5) , (- 2, 0) and (3 , 1) are the vertices of a rhombus.
A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.
Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.
Find distance between point A(– 3, 4) and origin O
Show that the points (2, 0), (– 2, 0) and (0, 2) are vertices of a triangle. State the type of triangle with reason
Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.
If the distance between the points (4, P) and (1, 0) is 5, then the value of p 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 ______.
