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

Video Transcript

Determine the length of the perpendicular from a point 𝐴 zero, zero to the line 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero.

We don’t actually need a graph to answer this question. But we can go ahead and make a sketch. Here’s a sketch. We have point 𝐴 at our origin. If this is the line 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, then we could represent the perpendicular by this pink dotted line. And that’s the length we’re trying to determine.

And we know something about the perpendicular distance from a point to a line. Given the point 𝑚, 𝑛 and 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero, the distance will be equal to the absolute value of 𝐴 of 𝑚 plus 𝐵 of 𝑛 plus 𝑐 over the square root of 𝐴 squared plus 𝐵 squared. We have point zero, zero as our start. And when we substitute what we know, we get the distance is equal to the absolute value of 𝑎 times zero plus 𝑏 times zero plus 𝑐 over the square root of 𝑎 squared plus 𝑏 squared. We know that 𝑎 times zero plus 𝑏 times zero equals zero.

So we’ll have the absolute value of 𝑐 divided by the square root of 𝑎 squared plus 𝑏 squared.