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CCSS Math: 8.G.B.7

The main mast of a fishing boat
is supported by a sturdy rope that extends from the top
of the mast to the deck. If the mast is 20 feet tall and
the rope attaches to the deck 15 feet away from
the base of the mast, how long is the rope? So let's draw ourselves a boat
and make sure we understand what the deck and the mast
and all of that is. So let me draw a boat. I'll start with yellow. So let's say that
this is my boat. That is the deck of the boat. And the boat might look
something like this. It's a sailing boat. This is the water down here. And then the mast is the thing
that holds up the sail. So let me draw ourselves
a mast. And they say the mast
is 20 feet tall. So this distance right
here is 20 feet. That is what is holding
up the sail. I can draw it as a pole so it's
a little bit clearer. Even shade it in if we like. And then they say a rope
attaches to the deck 15 feet away from the base of the mast.
So this is the base of the mast. This is the
deck right here. The rope attaches 15 feet away
from the base of the mast. So if this is the base of the
mast, we go 15 feet, might be about that distance
right there. Let me mark that. And the rope attaches
right here. From the top of the mast
all the way that base. So the rope goes like that. And then they ask us, how
long is the rope? So there's a few things
you might realize. We're dealing with
a triangle here. And it's not any triangle. We're assuming that the mast
goes straight up and that the deck is straight
left and right. So this is a right triangle. This is a 90 degree
angle right here. And we know that, if we know two
sides of a right triangle, we can always figure out the
third side of a right triangle using the Pythagorean theorem. And all that tells us is it the
sum of the squares of the shorter sides of the triangle
are going to be equal to the square of the longer side. And that longer side is
call the hypotenuse. And in all cases, the hypotenuse
is the side opposite the 90 degree angle. It is always going to
be the longest side of our right triangle. So we need to figure out
the hypotenuse here. We know the lengths of the
two shorter sides. So we can see that if we take 15
squared, that's one of the short sides, I'm squaring it. And then add that to the square
of the other shorter side, to 20 feet squared. And when I say the shorter side,
I mean relative to the hypotenuse. The hypotenuse will always
be the longest side. Let's say the hypotenuse is in
green just so we get our color coding nice. That is going to be equal
to the rope squared. Or the length of the rope. Let's call this distance
right here r. r for rope. So 15 squared plus 20
squared is going to be equal to r squared. And what's 15 squared? It's 225. 20 squared is 400. And that's going to be
equal to r squared. Now 225 plus 400 is 625. 625 is equal to r squared. And then we can take the
principal root of both sides of this equation. Because we're talking about
distances, we want the positive square root. So you take the positive square
root, or the principal root, of both sides
of this equation. And you are left with
r is equal to the square root of 625. You can play with it a little
bit if you like. But if you've ever played with
numbers around 25, you'll see that this is 25 squared. So r is equal to the square
root of 625, which is 25. So this distance right here,
the length of the rope, is equal to 25 feet.