8th grade (Illustrative Mathematics)
Course: 8th grade (Illustrative Mathematics) > Unit 8Lesson 10: Lesson 10: Applications of the Pythagorean Theorem
Pythagorean theorem word problem: fishing boat
If we assume that the mast of a boat is at a right angle to the deck, then we can model the length of a rope stretched between the mast and deck as the hypotenuse of a right triangle. Then we can use the Pythagorean theorem to relate the lengths. Created by Sal Khan and Monterey Institute for Technology and Education.
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- Does the Pythagorean theorem only work on right triangles?(55 votes)
- Good question!
Draw a right-angled triangle, with a square drawn on each of the sides. The areas of the two smaller squares, added together, is equal to the area of the largest square.
Now draw two more triangles. In the first one, keep the longest side the same, but make the two shorter sides a bit shorter than in your first triangle. They will meet at an angle greater than 90°. Can you see that the areas of the squares drawn on their sides will add up to less than before? In the next triangle, keep the longest side the same, but make the two shorter sides a bit longer than in your first triangle. They will meet at an angle less than 90°. Can you see that the areas of the squares drawn on their sides will add up to more than before?
It is only when the two shorter sides meet at 90° do you get the relationship between the areas of the squares.
See also http://wikids-math.wikispaces.com/Pythagoras%27+Theorem(6 votes)
- Does it matter what order you pit the a2 or b2(11 votes)
- In the equation, no it does not because the terms are being added together. This is called the commutative property. In math we write the terms in alphabetical order according to the letter. That is why you see a2 written first, because a is before b in the alphabet.
On the triangle, as long as you are using the smaller sides of the triangle, called legs, it does not matter which you label as a or b.(12 votes)
- What if the mast isn't absolutely upright? How would we find the length of the rope then?(4 votes)
- If it isn't a right triangle you will need to use the Law of Cosines, or the Law of Sines.(12 votes)
- how is the Pythagorean useful in word problems?(7 votes)
- It you need to find the length of a ladder on the side of a house, which makes a right triangle(3 votes)
- Is there another theorem for other types of triangles?(6 votes)
- There are several theorems relating the sides and angles of general triangles. Try the Law of Sines and Law of Cosines in the Trigonometry section.(5 votes)
- can someone help? i have a hard time converting the answers in the modules.(6 votes)
- Try finding the lengths first before you calculate anything and make sure to draw a triangle so it makes it alot easier for yourself!(2 votes)
- This might have been asked before, but i couldn't find it.
Why can't you just (15^2)^1/2 + (20^2)^1/2= (c^2)^1/2 ---> 15+20=35. Why doesn't this work?(2 votes)
- This doesn't work because you have to take the square root of the whole left side to undo the "c^2" on the right side and get c by itself. Consider this example:
Let's say we had a right triangle with lengths 3, 4, and 5. We would write the pythagorean theorem like this:
3^2 + 4^2 = 5^2
And this works, because 3^2 + 4^2 is the same as 9 + 16, which is 25, which is 5^2. So the equation works.
However, what would happen if I tried to solve it the way you did in your question? It would look like this:
(3^2)^1/2 + (4^2)^1/2 = (5^2)^1/2
When we take a number and square it, and then take the square root of that, we just get the original number again. So "(3^2)^1/2" would just be "3", and "(4^2)^1/2" would just be 4, and "(5^2)^1/2" would just be 5. So we could re-write the equation like this:
3 + 4 = 5
But this clearly isn't true.
The reason it didn't work is because square roots do not distribute. This means that if you take the square root of both sides to get rid of the "^2" on the "c", you have to take the square root of the WHOLE left side too, not just the individual terms. So the actual equation would be:
a^2 + b^2 = c^2
(a^2 + b^2) ^ 1/2 = c
You can't just throw that "1/2" onto the a^2 and the b^2. That would get you a totally different answer.
I hope this helped.(8 votes)
- Is there any theorems for the other types of triangles (scalene, equalateral) like there is the Pythagorean theorem?(2 votes)
- Most of the times no but in some unique cases yes(1 vote)
- is Pythagoras theorem is the root of trigonometry?
why is there a need for Pythagoras theorem?(3 votes)
- In a sense, creation of the unit circle is based on the Pythagorean theorem, but it you have a right angle and the measure of two sides without knowing the angles, the Pythagorean theorem is the quickest way of finding the third side (and most accurate) without having to go through trig functions to find angles.(4 votes)
- Why is the answer given as 6√3 instead of 10.4? It requires that you round it, but I was just wondering if, on a test for instance, one is a 'better' answer than the other.(3 votes)
- It is better to leave it in the simpler format of 6 square roots of 3 unless it asks you to simplify it.(4 votes)
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.