Determine whether the given coordinates are the vertices of a triangle. Explain. R(1,-4), S(-3,-

Determine whether the given coordinates are the vertices of...

Key Concepts

Determinant (Area) Method

The determinant method involves calculating the area of the triangle formed by three points using a specific formula. If the computed area equals zero, it implies that the points are collinear because they fail to enclose any area, meaning they cannot form a triangle.

Slope Method

The slope method is a technique used to test for collinearity by calculating the slopes between pairs of points. If the slopes between two different pairs of points are equal, this indicates the points lie along the same line, confirming collinearity.

Collinearity

Collinearity is the property where three or more points lie on the same straight line. In coordinate geometry, this is an important check when determining if points can form a triangle, because if the three points are collinear, they do not enclose an area and therefore cannot be the vertices of a triangle.

Transcript

00:02 To determine whether these three vertices will form a triangle, we first need to find the distance between each pair of vertices that will translate to the side lengths.

00:13 And then once we have those distances, we can compare the triangle inequality theorem for the three inequalities to make sure that the sum of any two sides is greater than the third side.

00:24 So starting with finding our distance between the pairs of vertices, we're going to use our distance formula that i have put up here as a reminder.

00:38 And using vertices rs to find the length of the segment, we have rs is equal to the square root of negative 3 minus 1 squared plus negative 20 minus negative 4 or negative 20 plus 4 squared.

00:55 And then we're just going to simplify it.

00:57 And this is going to be the square root of negative 4 squared plus negative 16 squared, which is equal to the square root of 16 plus 256, which finally simplifies to the square root of 272.

01:14 I'm going to leave everything in radical form so that we don't have any rounding errors.

01:19 Let's move on to the segment formed by vertices st.

01:23 So st is going to be equal to the square root of 5 minus negative 3 or 5 plus 3 squared plus 12 minus negative 20, 12 plus 20 squared, which is equal to the square root of 8 squared plus 32 squared, which is equal to the square root 64 plus 32 squared is 1024.

01:51 And that finally simplifies to the square root of 1088.

01:58 Just one more to go.

02:01 And we need the segment length of rt.

02:06 So rt is equal to the square root of 5 minus 1 squared plus 12 minus negative 4, which is 12 plus 4 squared, simplifies to the square root of 4 squared plus 16 squared, which is equal to the square of 16 plus 256, which is equal to the square root of 272.

02:36 It appears we have two side lengths that are equal.

02:41 We're still going to have to see if our triangle inequality will hold.

02:47 So to form a triangle, remember we have to show that the sum of two short sides is going to be greater than the third side.

02:53 Any two short sides or any two of the sides will be greater than the third side...

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