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प्रश्न
By using the distance formula prove that each of the following sets of points are the vertices of a right angled triangle.
(i) (6, 2), (3, -1) and (- 2, 4)
(ii) (-2, 2), (8, -2) and (-4, -3).
उत्तर
Let A(6, 2), B(3, -1) and C(-2, 4) be the given points
AB = `sqrt((6 - 3)^2 + (2 + 1)^2)`
= `sqrt(9 + 9) = sqrt(18) = 3sqrt(2)"units"`.
BC = `sqrt((3 + 2)^2 + (-1 -4)^2)`
= `sqrt(25 + 25) = 5sqrt(2)"units"`.
AC = `sqrt((6 + 2)^2 + (2 - 4)^2)`
= `sqrt(64 + 4)`
= `sqrt(68)"units"`.
Now AB2 + BC2 = `(3sqrt(2))^2 + (5sqrt(2))^2`
= 18 + 50 = 68 = AC2
AB2 + BC2 = AC2
Hence, the triangle is right angled at A.
Hence proved.
(ii) Let A (-2, 2), B(8, -2) and C(-4, -3) be the given points
AB = `sqrt((8 + 2)^2 + (-2 -2)^2)`
= `sqrt(116)"units"`.
BC = `sqrt((8 + 4)^2 + (-2 + 3)^2)`
= `sqrt(145)"units"`.
AC = `sqrt((-2 + 4)^2 + (2 + 3)^2)`
= `sqrt(29)"units"`.
Now AB2 + AC2
= 116 + 29
= 145
= BC2
Since AB + AC2 = BC2
Hence, the triangle is right angled at A.
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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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- 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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