Question Video: Determining the Relation between Two Straight Lines given the Coordinates of the Points Lying on Them Mathematics

Video Transcript

Given that the coordinates of the points 𝐴, 𝐵, 𝐶, and 𝐷 are negative 15, eight; negative six, 10; negative eight, negative seven; and negative six, negative 16, respectively, determine whether line 𝐴𝐵 and line 𝐶𝐷 are parallel, perpendicular, or neither.

Points 𝐴 and 𝐵 fall on the line 𝐴𝐵 and points 𝐶 and 𝐷 fall on the line 𝐶𝐷. To classify the lines, we have to remember: parallel lines have the same slope and they do not intersect. Perpendicular lines have negative reciprocal slopes and intersect at a 90-degree angle. And neither are lines that are not parallel or perpendicular, lines that do intersect but do not form a right angle. This means to consider whether or not these lines are parallel or perpendicular, we need to know the slopes of these lines.

In the general form, 𝑦 equals 𝑚𝑥 plus 𝑏, the 𝑚 represents the slope. And we can find the slope 𝑚 if we have two points by saying 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. In order to categorize these lines, we need to find the slopes of line 𝐴𝐵 and line 𝐶𝐷. We can start with line 𝐴𝐵. Let point 𝐴 be 𝑥 one, 𝑦 one and point 𝐵 be 𝑥 two, 𝑦 two. Then, the slope will be 10 minus eight over negative six minus negative 15. 10 minus eight is two. Negative six minus negative 15 is negative six plus 15, which is positive nine. So, we can say that the slope of line 𝐴𝐵 is two-ninths.

We repeat this process for line 𝐶𝐷. Let 𝐶 be 𝑥 one, 𝑦 one and 𝐷 be 𝑥 two, 𝑦 two. And we’ll get 𝑚 equals negative 16 minus negative seven over negative six minus negative eight. Negative 16 minus negative seven is negative 16 plus seven, which is negative nine. Negative six minus negative eight is negative six plus eight which is two. The slope of line 𝐶𝐷 is then negative nine over two.

If we compared these two slopes, negative nine over two is the negative reciprocal of two over nine. And if you weren’t sure, you can multiply them together. Reciprocals multiply together to equal one and negative reciprocals multiply together to equal negative one. These two slopes are the negative reciprocals of one another, making these lines perpendicular.