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प्रश्न
What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?
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उत्तर
The points are A(2, –2), B(7, 3), C(11, –1) and D(6, –6)
Using distance formula,
d = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
AB = `sqrt((7 - 2)^2 + (3 + 2)^2`
= `sqrt((5)^2 + (5)^2`
= `sqrt(25 + 25)`
= `sqrt(50)`
= 5`sqrt(2)`
BC = `sqrt((11 - 7)^2 + (-1 - 3)^2`
= `sqrt((4)^2 + (-4)^2`
= `sqrt(16 + 16)`
= `sqrt(32)`
= `4sqrt(2)`
CD = `sqrt((6 - 11)^2 + (-6 + 1)^2`
= `sqrt((-5)^2 + (-5)^2`
= `sqrt(25 + 25)`
= `sqrt(50)`
= `5sqrt(2)`
DA = `sqrt((2 - 6)^2 + (-2 + 6)^2`
= `sqrt((-4)^2 + (4)^2`
= `sqrt(16 + 16)`
= `sqrt(32)`
= `4sqrt(2)`
Finding diagonals AC and BD, we get,
AC = `sqrt((11 - 2)^2 + (-1 + 2)^2`
= `sqrt((9)^2 + (1)^2`
= `sqrt(81 + 1)`
= `sqrt(82)`
And BD = `sqrt((6 - 7)^2 + (-6 - 3)^2`
= `sqrt((-1)^2 + (-9)^2`
= `sqrt(1 + 81)`
= `sqrt(82)`
The quadrilateral formed is rectangle.
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संबंधित प्रश्न
Find the co-ordinates of points of trisection of the line segment joining the point (6, –9) and the origin.
Using the distance formula, show that the given points are collinear:
(-2, 5), (0,1) and (2, -3)
Find the distance between the following pairs of point in the coordinate plane :
(7 , -7) and (2 , 5)
Find the distance of the following point from the origin :
(6 , 8)
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.
Prove that the points (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Find the distance between the origin and the point:
(-8, 6)
Find the distance between the origin and the point:
(-5, -12)
Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.
Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square ABCD.
Find the distance of the following points from origin.
(5, 6)
Find distance between point A(– 3, 4) and origin O
Show that P(– 2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle
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 ______.
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 ______.
The distance of the point P(–6, 8) from the origin is ______.
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
The points A(–1, –2), B(4, 3), C(2, 5) and D(–3, 0) in that order form a rectangle.
