Question Video: Identifying the Distance between a Given Point and a Straight Line Mathematics

Video Transcript

Determine the length of the perpendicular from a point 𝐴: 𝑥 one, 𝑦 one to the line 𝑦 equals zero.

At first, it might not seem like we have enough information to solve this problem, but let’s sketch a coordinate grid. We have our 𝑥-axis and our 𝑦-axis. We want to know the perpendicular from the point 𝑥 one, 𝑦 one. But we can’t graph 𝑥 one, 𝑦 one. However, we can graph the line 𝑦 equals zero. The line 𝑦 equals zero is the 𝑥-axis. If we put 𝑥 one, 𝑦 one somewhere in the first quadrant, the perpendicular from this point to the line 𝑦 equals zero is going to be a vertical line. But what would the length of that vertical line be? It will be the distance of that point from the 𝑥-axis. And that’s going to be its 𝑦-coordinate. This coordinate is 𝑦 one units away from the 𝑥-axis.

However, we need to consider one other case. What if this is our point 𝑥 one, 𝑦 one? The length of this perpendicular segment is still going to be a vertical line. However, in this case, 𝑦 sub one is negative, and we can’t have negative distance. In this case, we would have to say that the distance from our point to the line 𝑦 equals zero is the absolute value of 𝑦 sub one. And so if we want to accurately describe the length from our point to the line 𝑦 equals zero, we need to say it’s going to be the absolute value of 𝑦 sub one. This will be true in every case no matter if 𝑦 sub one is positive or negative.