Question Video: Determining the Relation between Two Given Lines | Nagwa Question Video: Determining the Relation between Two Given Lines | Nagwa

Question Video: Determining the Relation between Two Given Lines Mathematics • Third Year of Preparatory School

Let 𝐿 be the line through the points (−7, −7) and (−9, 6) and 𝑀 the line through (1, 1) and (14, 3). Which of the following is true about the lines 𝐿 and 𝑀? [A] They are parallel. [B] They are perpendicular. [C] They are intersecting but not perpendicular.

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Video Transcript

Let 𝐿 be the line through the points negative seven, negative seven and negative nine, six and 𝑀 the line through one, one and 14, three. Which of the following is true about the lines 𝐿 and 𝑀? Option (A) they are parallel, option (B) they are perpendicular, or option (C) they are intersecting but not perpendicular.

It might be worthwhile beginning this question with a quick sketch of the two lines through the two sets of points. When we do this, we can observe that the two lines do in fact intersect. We can therefore say that these two lines are not parallel, so we can eliminate option (A). Now, we can remember that two lines are perpendicular if they intersect or meet at right angles. From the diagram, it does appear that the two lines are at right angles. But it could be the case that the two lines are nearly perpendicular and it’s not possible to distinguish this from the diagram. Generally, it’s not a very good idea to just use a sketch to determine if lines are parallel or perpendicular. In fact, we should perform some sort of calculation.

We can recall that if two straight lines have slopes of 𝑚 sub one and 𝑚 sub two, then they are perpendicular if 𝑚 sub two is equal to negative one over 𝑚 sub one. We’ll first need to calculate the slopes of each of the lines 𝐿 and 𝑀. The slope of the line passing through two points with coordinates 𝑥 sub zero, 𝑦 sub zero and 𝑥 sub one, 𝑦 sub one is calculated as the slope 𝑚 is equal to 𝑦 sub one minus 𝑦 sub zero over 𝑥 sub one minus 𝑥 sub zero. For line 𝐿 then, its slope 𝑚 sub one is equal to six minus negative seven over negative nine minus negative seven, which simplifies to negative 13 over two.

Now, let’s find the slope of the line 𝑀. Its slope 𝑚 sub two will be calculated as three minus one over 14 minus one, and this is equal to two over 13. Now, we can check if 𝑚 sub two is equal to negative one over 𝑚 sub one. If we didn’t know the value of 𝑚 sub two, we could find a perpendicular line to the line 𝐿 by taking 𝑚 sub two and setting it equal to negative one over negative 13 over two. And this would indeed give us a value of two thirteenths for 𝑚 sub two. We can therefore give the answer that the statement which is true about the lines 𝐿 and 𝑀 is option (B). They are perpendicular.

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