Given points A, B, and C . Find A B, B C, and A C . Are A, B, and C collinear? If so, which poin

Given points A, B, and C . Find A B, B C, and A C . Are A,...

Key Concepts

Slope

Slope is a measure of the inclination or steepness of a line, calculated as the change in the vertical direction divided by the change in the horizontal direction between two points. Equal slopes between pairs of points suggest the points are aligned linearly.

Identifying the Middle Point

In the context of collinear points, the middle point is the one where the sum of the distances from it to each of the other two equals the distance between the two outer points. This property helps in determining the order of points along the line.

Distance Formula

The distance formula computes the length between two points in a Cartesian plane and is derived from the Pythagorean theorem. It involves calculating the square root of the sum of the squares of the differences between the corresponding coordinates of the points.

Collinearity

Collinearity is the property of three or more points lying on the same straight line. To determine if points are collinear, one common method is to check if the slopes between each pair of points are equal or to verify if the sum of the distances between two pairs of points equals the distance between the two outer points.

Transcript

00:03 Okay, in this question, we are asked to determine if these three points are collinear and also to find the distances between each pair of points.

00:12 So i've gone ahead and plotted the three points on an x, y axis to decide whether they're a collinear or not.

00:19 And as we track down from, say, a to b, go down, one over, one down, one over, one over, one down, one over, we do not.

00:31 See this would be here is a line that contains a and that does not pass through b and c.

00:38 So these points are not collinear or non -colonial we might say.

00:46 Now as far as finding the distances between them, we need the distance formula for that.

00:51 So let's say the distance formula is the square root of the difference between the x coordinates squared, plus the difference between the y coordinates squared.

01:04 And so let's see, we'll start by taking the distance between a and b.

01:10 So that distance is going to be the square root of, let's see, the difference between the x coordinates.

01:17 I'm going to take point x, or sorry, point a here to be the point x2, y2, and then i'll use point b as x1, y1.

01:28 So when i write this in the distance formula, it's going to look something like.

01:34 Let's see.

01:35 I'll have the 3 is the x coordinate for point a minus the negative 3, which is the x coordinate for point b.

01:44 That's squared plus now the difference in the y coordinates.

01:50 So that'll be 4 minus, let's see, in point b, that's 0.

01:55 And that will be squared also.

01:57 So if we simplify here underneath the square root, of course, 3 minus a negative 3, that's 3 plus 3.

02:05 So this is 6 squared plus 4 minus 0.

02:09 That was before.

02:10 So 4 squared.

02:11 So this will be the square root of 36 plus 16, or the square root of 52 would be the distance between a and b.

02:25 Now let's go to the distance between b and c.

02:30 And since we already have set it up to have b, b point x1, y, 1, we'll go ahead and let c be x2y2 in this instance...

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