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प्रश्न
Determine whether the points are collinear.
L(–2, 3), M(1, –3), N(5, 4)
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उत्तर
L(–2, 3), M(1, –3), N(5, 4)
According to distance formula,
d(L, M) = `sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)`
d(L, M) = `sqrt([1 – (–2)]^2 + (–3 – 3)^2)`
d(L, M) = `sqrt((1 + 2)^2 + (–3 – 3)^2)`
d(L, M) = `sqrt((3)^2 + (–6)^2)`
d(L, M) = `sqrt(9 + 36)`
d(L, M) = `sqrt(45)`
d(L, M) = `sqrt(9 × 5)`
∴ d(L, M) = `3sqrt(5)` ...(1)
d(M, N) = `sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)`
d(M, N) = `sqrt((5 – 1)^2 + [4 – (– 3)]^2)`
d(M, N) = `sqrt((5 – 1)^2 + (4 + 3)^2)`
d(M, N) = `sqrt((4)^2 + (7)^2)`
d(M, N) = `sqrt(16 + 49)`
∴ d(M, N) = `sqrt(65)` ...(2)
d(L, N) = `sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)`
d(L, N) = `sqrt([5 – (– 2)]^2 + (4 – 3)^2)`
d(L, N) = `sqrt((5 + 2)^2 + (4 – 3)^2)`
d(L, N) = `sqrt((7)^2 + (1)^2)`
d(L, N) = `sqrt(49 + 1)`
d(L, N) = `sqrt(50)`
d(L, N) = `sqrt(25 × 2)`
∴ d(L, N) = `5sqrt(2)` ...(3)
From (1), (2), and (3),
Sum of two sides is not equal to the third side.
Hence, the given points are not collinear.
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संबंधित प्रश्न
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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:
The point on x axis equidistant from I and E is ______.
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Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
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