Pythagorean theorem application
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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.