Determine whether the following points are collinear A(-4, 4), K(-2, 5 2 ), N(4, -2)
Determine whether the following points are collinear A(-4, 4), K(-2, 5 2 ), N(4, -2)
A(–4, 4), \[K\left( - 2, \frac{5}{2} \right),\] N (4, –2)
\[\text { Slope of AK } = \frac{\frac{5}{2} - 4}{- 2 - \left( - 4 \right)} = \frac{\frac{- 3}{2}}{2} = \frac{- 3}{4}\]
\[\text { Slope of KN } = \frac{- 2 - \frac{5}{2}}{4 - \left( - 2 \right)} = \frac{- 3}{4}\]
Slope of AK=Slope of KN
Thus, the given points are collinear.
Explanation:-
The problem gives us three points in a 2-dimensional space, namely A(-4, 4), K(-2, 5/2), and N(4, -2). The problem asks us to determine whether these points are collinear.
To determine whether the points are collinear, we need to check whether the slopes of the lines connecting them are equal. We can use the formula for slope to find the slopes of the lines connecting A and K, and K and N:
slope of AK = (y-coordinate of K – y-coordinate of A) / (x-coordinate of K – x-coordinate of A)
slope of KN = (y-coordinate of N – y-coordinate of K) / (x-coordinate of N – x-coordinate of K)
We can simplify these formulas using the coordinates given in the problem:
slope of AK = (5/2 – 4) / (-2 – (-4)) = (-3/2) / 2 = -3/4
slope of KN = (-2 – 5/2) / (4 – (-2)) = (-3/2) / 6 = -1/4
Since the slopes of AK and KN are not equal, the points A, K, and N are not collinear. Therefore, there is no line that passes through all three points.
Chapter 5. Co-ordinate Geometry – Practice Set 5.3 (Page 121)