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

Determine the length of the perpendicular from a point 𝐴(0, 0) to the line π‘Žπ‘₯ + 𝑏𝑦 + 𝑐 = 0.

01:29

Video Transcript

Determine the length of the perpendicular from a point 𝐴 zero, zero to the line π‘Žπ‘₯ 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 π‘Žπ‘₯ 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 π‘š, 𝑛 and π‘Žπ‘₯ plus 𝑏𝑦 plus 𝑐 equals zero, the distance will be equal to the absolute value of 𝐴 of π‘š plus 𝐡 of 𝑛 plus 𝑐 over the square root of 𝐴 squared plus 𝐡 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 π‘Ž times zero plus 𝑏 times zero plus 𝑐 over the square root of π‘Ž squared plus 𝑏 squared. We know that π‘Ž times zero plus 𝑏 times zero equals zero.

So we’ll have the absolute value of 𝑐 divided by the square root of π‘Ž squared plus 𝑏 squared.

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