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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's 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 and then we can just use the Pythagorean theorem let me show want to 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 and 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 and 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 could see it on this grid here this is equal to 2 it's exactly 2 spaces and you can even think about it in terms of coordinates the coordinate of this point up here is negative 5 comma 8 negative 5 comma 8 the coordinate here is X is 4 y 6 4 comma 6 and so the coordinate over here is going to have the same the same y-coordinate is this point so it's going to be comma 6 it's going to have the same x-coordinate as this point so it's going to be negative 5 comma 6 and so notice you have a you're only changing 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 in the x-value we're going from negative 5 x equals negative 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 doreen 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 two squares let me do it over here to use the same red color 2 squared plus 9 squared plus 9 squared is going to be equal to our hypotenuse squared which I'm just calling C is going to be equal to C squared which is really the distance that's what we're trying to figure out so 2 squared that is 4 plus 9 squared is 81 that's going to be equal to C squared and so 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 or 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