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प्रश्न
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.
उत्तर
Circumcentre of Δ XYZ will pass through the vertices X , Y and Z
OX = OY (radi of same circle),
⇒ OX2 = OY2
(a - 1)2 + (b - 2)2 = (a - 3)2 + (b + 4)2
⇒ 1 - 2a + 4 - 4b = 9 - 6a + 16 + 8b
⇒ 4a - 12b = 20
⇒ a - 3b = 5 .........(1)
OY = OZ (radii of same circle)
OY2 = OZ2
(a - 3)2 + (b + 4)2 = (a - 5)2 + (b + 6)2
⇒ 9 - 6a + 16 + 8b = 25 - 10a + 36 + 12b
⇒ 4a - 4b = 36
⇒ a - b = 9 .........(2)
a - 3b = 5 ..........(1)
a - b = 9
- 2b = -4
b = 2
a = 11
Circumcentre of Δ XYZ is O (11 , 2)
Circumradius = `sqrt ((11 - 1)^2 + (2 - 2)^2) = sqrt 100` = 10 units
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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.
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