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प्रश्न
Find the circumcenter of the triangle whose vertices are (-2, -3), (-1, 0), (7, -6).
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उत्तर
The circumference of a triangle is equidistance from the vertices of a triangle.
Let A(-2, -3), B(-1, 0) and C(7, -6) vertices of the given triangle and let P(x,y) be the circumference of this triangle, Then
PA = PB = PC
Now, PA = PB
`=> sqrt((-2-x)^2 + (-3 -y)^2) = sqrt((-1 - x)^2 + (0 - y)^2`
`=> 4 + x^2 + 4x + 9 + y^2 + 6y = 1 + x^2 + 2xz + y^2`
`=> 4 + x^2 + 4x + 9 + y^2 + 6y - 1 - x^2 - 2x - y^2 = 0`
`=> 2x + 6y + 12 = 0
=> 2(x + 3y + 6) = 0
=> x + 3y + 6 = 0 ... eq (1)
And PB = PC
`=> sqrt((-1-x)^2 + (0 - y)^2) = sqrt((7- x)^2 + (-6 - y)^2)`
Squaring both the sides
`=> (-1 - x)^2 + y^2 = (7 - x)^2 + (-6 -y)^2`
`=> 1 + x^2 + 2x + y^2 = (7 - x)^2 + (-6 - y)^2`
`=> 1 + x^2 + 2x + y^2 - 49 - x^2 + 14x - 36 - y^2 - 12y`
=> 16x - 12y - 84 = 0
`=> 4(4x - 3y - 21) = 0`
=> 4x - 3y - 21 = 0 ......eq(2)
Adding eq(1) and eq(2)
`=> x + 3y + 6 + 4x - 3y - 21 = 0`
=> x + 3y + 6 + 4x -3y - 21 = 0
=> 5x - 15 = 0
=> x = 15/5`
=> x = 3
putting the value of x in eq (2) and
we get
`=> 4 x 3 - 3y - 21 = 0`
`=> 12 - 3y - 21 = 0`
`=> -3y - 9 = 0`
`=> y = (-9)/3 = -3`
So the coordinates of the circumcentre P are (3, -3)
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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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- 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:
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