Pythagorean Triangles and Triples
Use the Pythagorean Theorem to find the unknown side of a right triangle. You can see this illustrated below in the same right triangle. Note that the. The picture below shows the formula for the Pythagorean theorem. In the pictures below, side When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above. Look at the . Problem 3. Use the . Although Pythagoras is credited with the famous theorem, it is likely that the is called a Pythagorean triple. Some Pythagorean Triples: x y z 3 4 5 5 12
And we could take the positive square root of both sides. I guess, just if you look at it mathematically, it could be negative 5 as well. But we're dealing with distances, so we only care about the positive roots. So you take the principal root of both sides and you get 5 is equal to C. Or, the length of the longest side is equal to 5. Now, you can use the Pythagorean theorem, if we give you two of the sides, to figure out the third side no matter what the third side is.
So let's do another one right over here. Let's say that our triangle looks like this. And that is our right angle. Let's say this side over here has length 12, and let's say that this side over here has length 6. And we want to figure out this length right over there. Now, like I said, the first thing you want to do is identify the hypotenuse.
And that's going to be the side opposite the right angle. We have the right angle here. You go opposite the right angle. The longest side, the hypotenuse, is right there. So if we think about the Pythagorean theorem-- that A squared plus B squared is equal to C squared-- 12 you could view as C. This is the hypotenuse.
The C squared is the hypotenuse squared. So you could say 12 is equal to C. And then we could say that these sides, it doesn't matter whether you call one of them A or one of them B.
So let's just call this side right here. Let's say A is equal to 6. And then we say B-- this colored B-- is equal to question mark. And now we can apply the Pythagorean theorem. A squared, which is 6 squared, plus the unknown B squared is equal to the hypotenuse squared-- is equal to C squared. Is equal to 12 squared. And now we can solve for B. And notice the difference here.
Pythagorean theorem - Wikipedia
Now we're not solving for the hypotenuse. We're solving for one of the shorter sides. In the last example we solved for the hypotenuse. We solved for C. So that's why it's always important to recognize that A squared plus B squared plus C squared, C is the length of the hypotenuse.
So let's just solve for B here. So we get 6 squared is 36, plus B squared, is equal to 12 squared-- this 12 times is Now we can subtract 36 from both sides of this equation. On the left-hand side we're left with just a B squared is equal to-- now minus 36 is what?
And then you subtract 6, is So this is going to be So that's what B squared is, and now we want to take the principal root, or the positive root, of both sides. And you get B is equal to the square root, the principal root, of Now let's see if we can simplify this a little bit. The square root of And what we could do is we could take the prime factorization of and see how we can simplify this radical.
So is the same thing as 2 times 54, which is the same thing as 2 times 27, which is the same thing as 3 times 9. So we have the square root of is the same thing as the square root of 2 times 2 times-- well actually, I'm not done. So it's 2 times 2 times 3 times 3 times 3. And so, we have a couple of perfect squares in here. Let me rewrite it a little bit neater.
And this is all an exercise in simplifying radicals that you will bump into a lot while doing the Pythagorean theorem, so it doesn't hurt to do it right here. So this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there. And this is the same thing. And, you know, you wouldn't have to do all of this on paper. You could do it in your head. So this is the square root of 36 times the square root of 3. The principal root of 36 is 6.
So this simplifies to 6 square roots of 3. So the length of B, you could write it as the square root ofor you could say it's equal to 6 times the square root of 3. This is 12, this is 6. And the square root of 3, well this is going to be a 1 point something something. So it's going to be a little bit larger than 6. The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
The area of a rectangle is equal to the product of two adjacent sides. The area of a square is equal to the product of two of its sides follows from 3. Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.Pythagorean Triples
The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate. Similarly for B, A, and H.
Rescaling the Pythagorean Theorem – BetterExplained
The triangles are shown in two arrangements, the first of which leaves two squares a2 and b2 uncovered, the second of which leaves square c2 uncovered. A second proof by rearrangement is given by the middle animation.
A large square is formed with area c2, from four identical right triangles with sides a, b and c, fitted around a small central square. Then two rectangles are formed with sides a and b by moving the triangles.
Combining the smaller square with these rectangles produces two squares of areas a2 and b2, which must have the same area as the initial large square. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse — or conversely the large square can be divided as shown into pieces that fill the other two.
This way of cutting one figure into pieces and rearranging them to get another figure is called dissection. This shows the area of the large square equals that of the two smaller ones. In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts.