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A bead artist drills a tiny hole from the top corner of a

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cube bead to the opposite bottom corner of that bead.

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It's a cube bead.

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Each edge of the bead is 1/2 centimeters.

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How long is the hole that she drilled?

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So if I'm interpreting this properly, let me draw that

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cube bead, which I will just draw as a

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cube, a big cube here.

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Nicer cubes have been drawn in the history of cube drawing,

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but I'll try my best. So let's see.

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This is my best shot at a freehand cube.

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I think that does the job reasonably well.

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If this cube was transparent, you could even imagine kind of

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that backside over there, and the base of the cube if this

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was a transparent cube.

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They tell us that each edge of this cube bead-- they call

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this a cube bead-- each edge of the cube bead is 1/2

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centimeter.

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So this is 1/2 centimeters, this is 1/2 centimeters, and

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that is 1/2 centimeters.

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And then they say that they drilled a hole from the top

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corner of a cube bead to the opposite bottom

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corner of the bead.

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From the top corner to the opposite bottom.

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So I think they're talking about the longest diagonal

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that can fit in the cube.

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Going from the back, right top corner, over here, to the

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front, left bottom corner, right over there, that's the

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longest diagonal that could fit.

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So let me draw that.

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So it would look something like this.

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This is the distance in question.

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How long is the hole that she drilled?

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That is the distance that we have to figure out.

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And let's see how we can think about that.

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And this is a bit of a classic in terms of

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figuring out a distance.

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The longest diagonal of a cube or some type of

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a rectangular prism.

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And the trick here is to see that this is the hypotenuse of

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a triangle, although the triangle isn't completely

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obvious to you just yet.

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And to think about this as the hypotenuse of a right

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triangle, what you have to do is visualize the diagonal of

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the bottom face of this cube.

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Think about this diagonal right here that I'm drawing.

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So the diagonal of the bottom face of this cube.

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If you think of it that way, then all of a sudden we have a

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right triangle that's kind of going across this cube.

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This is a right angle over here.

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We know what this length is.

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This is 1/2 centimeters.

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If we could figure out the length of this orange side

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right here, this part that's going diagonally across the

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base of the cube, then we could use the Pythagorean

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theorem to figure out what the distance in question is, this

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longest diagonal of the cube.

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Now, to figure out this orange side, you just have to

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visualize the base of this cube.

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Let's look at this base.

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If we were just in the cube looking straight down at the

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base, it would look something like this.

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You'd have this side-- let me color code them; it will make

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it easier to visualize I think-- you have this side

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right here, which I will do in this blue color.

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So I'll draw it over here.

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You have this side in that blue color.

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And then you have this side over here, which I'll do in

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the magenta color.

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You have the side over there in the magenta color.

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They're both of length 1/2, so I'll draw them with roughly

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the same length.

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And then you have this orange side that goes across to the

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bottom of the base of the cube.

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And I'll do it as the same orange dotted line right

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there, just like that.

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Let me extend this side a little bit.

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And we know this is a right angle.

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This is a cube we're dealing with, so

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this is a right angle.

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We know that this side is 1/2 centimeter.

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We know that this side right over here is 1/2 centimeter.

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So if we wanted to figure out this diagonal or this length

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right here, we can just use the Pythagorean theorem.

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We know that 1/2 squared-- if we called this, I don't know,

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if we called this right here x-- we know that x squared,

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the hypotenuse squared, is equal to the the sum of the

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squares of these two guys, is equal to 1/2

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squared plus 1/2 squared.

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Or if we want to solve for x, we could say x is equal to the

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square root-- what's this going to be?

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The square root of 1/4, That's 1/2 squared, plus 1/2 squared,

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which is 1/4.

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Or, what's 1/4 plus 1/4?

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So x is equal to the square root of 1/4 plus 1/4, that's

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2/4, or 1/2.

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Or you could write that as 1 over the square root of 2.

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Now, we could rationalize this, but I'll leave it like

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that for now.

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It'll keep things simple.

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So this distance right here is 1 over the square root of 2,

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or we could say this distance right over here is 1 over the

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square root of 2.

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So if we were to draw the right triangle where the

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hypotenuse is the length of that hole, let me draw that

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right triangle.

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So we have our hypotenuse, which is this

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long thing over here.

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Let me just draw it like this.

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That long thing over there.

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We have this side over here in yellow.

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This side right here.

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We have that side right there.

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It is of length 1/2 centimeters.

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And then we have the side we just figured out, this side

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right here, that x side.

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This right over here.

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This side over here, it is a right angle.

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We just figured out this length.

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It is 1 over the square root of 2 centimeters.

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Now, to figure out the length of the hole, we can apply the

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Pythagorean theorem again.

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This is, let me say h for hole.

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Length of hole.

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Actually, even better, let's say l.

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l for length.

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Then we can say that l squared is going to be equal to 1 over

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the square root of 2 squared-- we're just applying the

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Pythagorean theorem-- plus 1 squared.

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And once again, we know that this is the longest side.

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It's opposite the right angle.

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So it's going to be the sum of the squares of

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the other two sides.

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And so that's going to be equal to-- so we get l squared

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is equal to-- what's 1 over the square root of 2 squared?

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Well, 1 squared is 1, over the square root of 2 squared is 2.

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And then you're going to have plus 1/2 squared.

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1/2 squared is plus 1/4.

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So l squared is equal to 1/2 plus 1/4.

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1/2 plus 1/4 is what?

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Let's do it over here.

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1/2-- we can write it as 2/4-- plus 1/4 is equal to 3/4.

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So this right here is equal to 3/4.

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So we have l squared-- I changed the shades of green

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just there-- is equal to 3 over 4, so l is equal to the

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square root of 3 over 4.

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So we take the square root of both sides, you get l is equal

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to the square root of 3 over the square root of 4.

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What's the square root of 4?

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The square root of 4 is 2.

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And we're done.

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The length of the hole that was drilled in this bead from

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this corner over here all the way over here is the square

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root of 3 over 2 centimeters.

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Because everything we've been dealing with has been

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centimeters.

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