Determine whether the points are collinear. L(-2, 3), M(1, -3), N(5, 4)
Determine whether the points are collinear. L(-2, 3), M(1, -3), N(5, 4)
L(–2, 3), M(1, –3), N(5, 4)
\[LM = \sqrt{\left( - 2 - 1 \right)^2 + \left( 3 - \left( - 3 \right) \right)^2}\]
\[ = \sqrt{\left( - 3 \right)^2 + 6^2} = \sqrt{9 + 36}\]
\[ = \sqrt{45} = 3\sqrt{5}\]
\[MN = \sqrt{\left( 1 - 5 \right)^2 + \left( - 3 - 4 \right)^2}\]
\[ = \sqrt{16 + 49}\]
\[ = \sqrt{65}\]
\[LN = \sqrt{\left( - 2 - 5 \right)^2 + \left( 3 - 4 \right)^2}\]
\[ = \sqrt{\left( - 7 \right)^2 + \left( - 1 \right)^2}\]
\[ = \sqrt{49 + 1} = \sqrt{50}\]
\[ = 5\sqrt{2}\]
Sum of two sides is not equal to the third side. Hence, the given points are not collinear.
Explanation:-
The given problem involves finding the distances between three points on a 2-dimensional plane and determining if they are collinear (i.e., if they lie on a straight line).
The three points are L(-2, 3), M(1, -3), and N(5, 4).
To find the distance between points L and M, we can use the distance formula:
LM = sqrt((x2 – x1)^2 + (y2 – y1)^2)
Plugging in the values for L(-2, 3) and M(1, -3):
LM = sqrt((1 – (-2))^2 + (-3 – 3)^2) = sqrt((3)^2 + (-6)^2) = sqrt(9 + 36) = sqrt(45) = 3 sqrt(5)
Similarly, to find the distance between points M and N:
MN = sqrt((x2 – x1)^2 + (y2 – y1)^2)
Plugging in the values for M(1, -3) and N(5, 4):
MN = sqrt((5 – 1)^2 + (4 – (-3))^2) = sqrt((4)^2 + (7)^2) = sqrt(16 + 49) = sqrt(65)
And to find the distance between points L and N:
LN = sqrt((x2 – x1)^2 + (y2 – y1)^2)
Plugging in the values for L(-2, 3) and N(5, 4):
LN = sqrt((5 – (-2))^2 + (4 – 3)^2) = sqrt((7)^2 + (1)^2) = sqrt(49 + 1) = sqrt(50) = 5 sqrt(2)
Now, we check if the points are collinear by seeing if the sum of any two sides is greater than or equal to the third side. If the points are collinear, then this condition should hold true for all three pairs of points.
For example, if the points were collinear, then we would expect:
LM + MN >= LN LM + LN >= MN MN + LN >= LM
However, in this case, we find:
3 sqrt(5) + sqrt(65) = 8.13 < 5 sqrt(2) 3 sqrt(5) + 5 sqrt(2) = 14.47 > sqrt(65) sqrt(65) + 5 sqrt(2) = 12.18 > 3 sqrt(5)
Thus, we can conclude that the given points are not collinear.
Chapter 5. Co-ordinate Geometry – Practice Set 5.1 (Page 107)