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- We are asked what is the distance

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between the following points.

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Pause this video and see[br]if you can figure it out.

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There's multiple ways to think about it.

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The way I think about it[br]is really to try to draw

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a right triangle where these points,

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where the line that connects[br]these points is the hypotenuse

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and then we can just use[br]the Pythagorean Theorem.

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Let me show you what I am talking about.

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Let me draw a right triangle, here.

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That is the height of my right triangle

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and this is the width[br]of my right triangle.

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Then the hypotenuse will[br]connect these two points.

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I could use my little[br]ruler tool here to connect

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that point and that[br]point right over there.

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I'll color it in orange.

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There you have it.

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There you have it.

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I have a right triangle[br]where the line that connects

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those two points is the[br]hypotenuse of that right triangle.

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Why is that useful?

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From this, can you pause[br]the video and figure out

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the length of that orange[br]line, which is the distance

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between those two points?

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What is the length of this red line?

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You could see it on this grid, here.

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This is equal to two.

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It's exactly two spaces, and[br]you could even think about it

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in terms of coordinates.

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The coordinate of this point up here

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is negative five comma eight.

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Negative five comma eight.

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The coordinate here is[br]X is four, Y is six.

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Four comma six, and so[br]the coordinate over here

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is going to have the same[br]Y coordinate as this point.

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This is going to be comma six.

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It's going to have the same[br]X coordinate as this point.

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This is going to be[br]negative five comma six.

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Notice, you're only[br]changing in the Y direction

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and you're changing by two.

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What's the length of this line?

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You could count it out, one, two, three,

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four, five, six, seven, eight, nine.

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It's nine, or you could even say hey look,

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we're only changing in the X value.

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We're going from negative five,

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X equals negative five, to X equals four.

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We're going to increase by nine.

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All of that just sets us up so that

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we can use the Pythagorean Theorem.

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If we call this C, we know[br]that A squared plus B squared

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is equal to C squared, or we[br]could say that two squared ...

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Let me do it over here.

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Use that same red color.

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Two squared plus nine[br]squared, plus nine squared,

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is going to be equal to[br]our hypotenuse square,

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which I'm just calling C, is[br]going to be equal to C squared,

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which is really the distance.

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That's what we're trying to figure out.

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Two squared, that is four,[br]plus nine squared is 81.

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That's going to be equal to C squared.

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We get C squared is equal to 85.

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C squared is equal to 85 or C

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is equal to the principal root of 85.

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Can I simplify that a little bit?

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Let's see.

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How many times does five go into 85?

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It goes, let's see, it goes 17 times.

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Neither of those are perfect squares.

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Yeah, that's 50 plus 35.

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Yeah, I think that's about[br]as simple as I can write it.

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If you wanted to express it as a decimal,

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you could approximate it by[br]putting this into a calculator

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and however precise you want[br]your approximation to be.

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That over here, that's[br]the length of this line,

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our hypotenuse and our right triangle,

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but more importantly for[br]the question they're asking,

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the distance between those points.