Example finding distance with Pythagorean theorem

We are asked what is the distance between the following points, so pause this video and see if you can figure it out. Well, there are multiple ways to think about it. The way I think about it is really to try to draw a right triangle where these points, where the line that connects these points, is the hypotenuse.

Then we can just use the Pythagorean theorem. Let me show you what I am talking about. So let me draw a right triangle here. So that is the height of my right triangle, and this is the width of my right triangle. Then, the hypotenuse will connect these two points.

I could use my little ruler tool here to connect that point and that point right over there. I'll color it in orange, so there you have it. I have a right triangle where the line that connects those two points is the hypotenuse of that right triangle.

Now, why is that useful? From this, can you pause the video and figure out the length of that orange line, which is the distance between those two points? Well, what is the length of this red line? Well, you can see it on this grid here. This is equal to 2; it's exactly 2 spaces.

You can even think about it in terms of coordinates. The coordinate of this point up here is negative 5, comma 8 (−5, 8). The coordinate here is x equals 4, y equals 6 (4, 6). So the coordinate over here is going to have the same y-coordinate as this point, so it's going to be negative 5, comma 6 (−5, 6).

Notice you have a change only in the y-direction, and you're changing by 2. Now, what's the length of this line? Well, you could count it out: 1, 2, 3, 4, 5, 6, 7, 8, 9. So it's 9. Or you could even say, hey look, we're only changing our x-value. We're going from x equals -5 to x equals 4, so we're going to increase by 9.

Now, all of that just sets us up so that we can use the Pythagorean theorem. If we call this C, well, we know that a squared plus b squared is equal to c squared, or we could say that 2 squared plus 9 squared is going to be equal to our hypotenuse squared, which I'm just calling C squared.

So 2 squared, that is 4, plus 9 squared is 81, and that's going to be equal to C squared. Thus, we get C squared is equal to 85. C squared is equal to 85, or C is equal to the principal root of 85.

Now, can I simplify that a bit? Let's see, how many times does 5 go into 85? It goes, let's see, it goes 17 times. So neither of those are perfect squares. Yeah, that's 50 plus 35. So yeah, I think that's about as simple as I can write it.

If you wanted to express it as a decimal, you could approximate it by putting this into a calculator and however precise you want your approximation to be. But that, over here, that's the length of this line, our hypotenuse in our right triangle.

But more importantly, for the question they're asking, the distance between those points.