Collinear Points Three or more points are collinear when they all lie on the same line. Use the

step 1 we are given two sets of triples of points. So one of these sets of points, which we'll call A,

Collinear Points Three or more points are collinear when...

Key Concepts

Distance Relationships

When assessing collinearity using distances, one key relationship is that for collinear points, the sum of the distances between two adjacent pairs will equal the distance between the outermost points. This distance additivity property arises because the points lie along a straight path. Verifying this relationship with computed distances can serve as a test for collinearity.

Graphical Analysis

Graphical analysis in the Cartesian plane involves plotting the points and visually inspecting their arrangement. This method provides an intuitive understanding of collinearity, as points lying on the same straight line will visually align. Combining graphical methods with algebraic techniques like the Distance Formula offers both a visual and quantitative means of confirming collinearity.

Collinearity

Collinearity refers to the condition when three or more points lie on a single straight line. It is a foundational concept in geometry and is critical for understanding the alignment and relationships between points in a set. Determining collinearity allows one to decide if points form a linear configuration, which is useful in proofs and many geometric applications.

Distance Formula

The Distance Formula is a tool derived from the Pythagorean theorem that calculates the straight-line distance between two points in a Cartesian plane. By measuring the horizontal and vertical separations between the two points, the formula provides the length of the segment connecting them. This calculation is essential for quantifying spatial relationships in analytical geometry.

Transcript

00:02 We are given two sets of triples of points.

00:12 So one of these sets of points, which we'll call a, b, and c, well, the points have coordinates to 3, b has coordinates 26, and c has coordinates 6 ,3, and our other set of points has points a, b and c with coordinates a, equals 8 3, b as coordinates 5, 2, and c as coordinates 2, and c as coordinates 2, 1.

01:31 Now, in part a, we're actually going to follow a sequence of steps here to prove whether or not these two sets of points are sets of co -linear points.

01:47 That is whether or not these sets of points each could or could not lie in a line together.

02:03 Now in part a, we're asked to use the distance formula to find the distances from a to b, from b to c, and from a to c.

02:13 So to do this, we'll use your distance formula.

02:18 So i'll break this up into two parts.

02:20 I'll call this set one, and this will be set two.

02:35 So for the first part of part a, we're just dealing with set 1.

02:42 Let's find the distance between a and b.

02:46 So this is distance between points 2 .3 and 2 .6.

02:56 This is, by the distance formula, the square root of 2 minus 2 squared plus 3 minus 6 squared, which simplifies to the square root of 0 squared plus negative 3 squared.

03:15 Or 9, which is the square root of 9 or 3.

03:22 We have the distance between b and c.

03:27 This is the distance between points 26 and 6 .3, and this is the square root of 2 minus 6 squared plus 6 minus 3 squared, or the square root of 4 squared, which is 16, plus 3 squared, which is 9, or the square root of 25, which is 5.

03:58 Finally, the distance from a to c is the distance from 0 .23 to the point 6.

04:07 Which is the square root of 2 minus 6 squared plus 3 minus 3 squared, which is the square root of 4 squared or 16 plus 0 squared, which is 0, or the square root of 16, which is 4 squared, which is 4.

04:25 So we have here that the distance between points a and b is 3, between points b and c is 5, and between points a and c is 4.

04:41 Now, we're asked to find a relationship among the distances for this set of points.

04:54 Well, just looking at numerical values, we see that 3 squared plus 4 squared is equal to let me make this a little bit simpler, actually.

05:37 Notice that if we add, say, 3 and 5, this is going to be 8, which we see is greater than 4.

05:47 If we add 3 and 4, this is 7, which we see is strictly greater than 5.

05:55 And finally, if we add 5 and 4, this is 9, which is strictly greater than 3.

06:02 So the relationship that exists among these distances for this set of points is that the sum of two distances is always greater than the third distance.

06:41 Now let's look at our second set of points, set two.

06:48 Here we have the distance from a to b is the distance from 0 .83 to 5 to.

06:55 This is the square root of 8 minus 5 squared plus 3 minus 2 squared, or the square root of 3 squared or 9 plus 1 squared or 1, which is the square root of 10.

07:16 We have the distance between b and c is the distance between 52 and 2 .1.

07:31 This is the square root of 5 minus 2 squared plus 2 minus 1 squared plus 2 minus 1 squared.

07:38 Which is the square root of 3 squared or 9 plus 1 squared or 1 which is the square root of 10.

07:48 And finally the distance from a to c is the distance from 0 .83 to the point to 1.

07:59 This is the square root of 8 minus 2 squared plus 3 minus 1 squared or the square root of 6 squared which is 36 plus 2 squared, which is 4.

08:17 So this is the square root of 40, which can also be written as 2 times the square root of 10.

08:33 Now let's look at the relationships among these distances.

08:38 Well, we see that the first distance, square root of 10, plus the second distance, square root of 10, is equal to 2 times the square root of 10, which is our third distance.

08:52 So already we see something different from the first set.

08:58 We have that the square of 10 plus the third distance, two times the square of 10 is equal to three times the square of 10, which is greater than the square root of 10, the remaining distance...

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