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प्रश्न
Determine whether the following point is collinear.
A(–4, 4), \[K\left( - 2, \frac{5}{2} \right),\] N (4, –2)
उत्तर
A(–4, 4), \[K\left( - 2, \frac{5}{2} \right),\] N (4, –2)
Slope of AK = `(5/2 - 4)/((-2) - (- 4))`
= `((5 - 8)/2)/((-2) + 4)`
= `((-3)/2)/2`
= `((-3)/2)/(2/1) = (-3)/2 xx 1/2 = (-3)/(2 xx 2)`
= `(-3)/4`
\[\text { Slope of KN } = \frac{(- 2) - \frac{5}{2}}{4 - \left( - 2 \right)}\]
= `(((-4) - 5)/2)/(4 + 2)`
= `((-9)/2)/(6/1) = (- 9)/2 xx 1/6 = (-9)/(6 xx 2) = (- 9)/12`
= `(- 3)/4`
Slope of AK = Slope of KN
Thus, the given points are collinear.
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Solution:
Slope of line = `("y"_2 - "y"_1)/("x"_2 - "x"_1)`
∴ Slope of line AB = `(2 - 1)/(8 - 6) = square` .......(i)
∴ Slope of line BC = `(4 - 2)/(9 - 8) = square` .....(ii)
∴ Slope of line CD = `(3 - 4)/(7 - 9) = square` .....(iii)
∴ Slope of line DA = `(3 - 1)/(7 - 6) = square` .....(iv)
∴ Slope of line AB = `square` ......[From (i) and (iii)]
∴ line AB || line CD
∴ Slope of line BC = `square` ......[From (ii) and (iv)]
∴ line BC || line DA
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∴ `square`ABCD is a parallelogram.
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