If three distinct points A, B, and C in a plane are such that the...

Question

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear.
(-1, -3), (-5, 12), (1, -11)

Accepted Answer

Determine if the given sets of points are collinear by calculating the slopes of the line segments formed by the points. And we notice here we have three different points, negative three, negative one, negative 7, 13 and four, negative eight. Now, to the time of the C co linear, we wanna see if the slopes of each segment are the same. So in other words, if we form a slope between these two points are they all equal to each other, we have three points, we can form three pairs of two points. Let's find ma B first for instance. Now let's say this is point A point B and point C where negative three, negative one is our first point A. We know our slope is Y two minus Y one over X two minus X one. I've seen point Bry two is 13. Ry one is negative one, X two is negative seven and X one is negative three. We then get native through or sorry, not native but 13 minus native 1, 13 plus one or 14 in the bottom negative seven plus three, negative four, which gives us negative 7/2. Let's do the same for NBC. So use point B and point C or our 2nd and 3rd points. Should we get minus Y one, N eight minus 13 divided by four minus negative seven on the top, we get near the 21 on the bottom. We get past of 11. That slope turns out to be negative 21/11. Finally, you will check M AC N eight minus negative one divided by or minus negative three. Give this negative eight plus one or negative seven divided by four plus 37 or negative one. Now, all three of these slopes are different from each other. We set, we have to have all three steps to be equal to each other for the set of points to be collinear. Since they are not equals, we can then say answer B these points are not collinear. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

If three distinct points A, B, and C in a plane are such that the slopes of nonvertical line segments AB, AC, and BC are equal, then A, B, and C are collinear. Otherwise, they are not. Use this fact to determine whether the three points given are collinear. (-1, -3), (-5, 12), (1, -11)

Key Concepts

Slope of a Line

Collinearity

Coordinate Geometry

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