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CCSS.Math:

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 let's draw ourselves a boat and make sure we understand what the deck and the mast and all that is so let me draw a boat let's see a nice color 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 so that is the deck of the boat this is the water down here that is the water and then the mast is the thing that holds up the sails let me draw ourselves a mast so the mast so let me draw the mast just like this that is my mast and they say if the mast is 20 feet tall it is 20 feet tall so this distance right here is 20 feet that distance right there that is what is holding up the sail I could 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 attach it 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 attached is 15 feet away from the base of the mast so the rope might attach so if this is the base of the mast we go 15 feet might be about that distance right there let me mark that this distance right there is 15 feet and the rope attaches right here from the top of the mast all the way to 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 this is a right triangle and we know that if we know two sides of a right triangle we can always figure out the third side of the right triangle using the Pythagorean theorem Pythagorean Pythagorean theorem and all that tells us is that the square 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 called the hypotenuse hypotenuse by a pot in use and in this kit well in all cases the hypotenuse is the side opposite the 90-degree angle 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 15 squared that's one of the shorter 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 that is going to be equal to 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 so let's call this distance right here are our 4 rope let me do a little bit neater than that our 4 rope is the length of the 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 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 could 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 could 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 this is 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