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The Pythagorean Theorem
Hello everyone, I'm Sergio Alvarez and today we'll be learning about the Pythagorean Theorem.
Now, before we jump into the Pythagorean Theorem, there are a couple of things we must know about this right triangle. The reason I know this is a right triangle is because of this little box. This little box serves as an indicator that this angle measures 90 degrees.
Let's learn something about the sides: A right triangle has one side that is longer than the other two.
The longest side of a right triangle can always be found across the ninety degree angle. This side is the longest side of my right triangle. This side it's called hypotenuse.
Notice that the two sides that create the ninety degree angle form a little L. L for Legs. These are the legs of the right triangle.
We will call this side leg a, and this side leg b, or vice versa.
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Ok, we are ready to learn the Pythagorean Theorem.
The Pythagorean Theorem says that in all right triangles, the sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse.
I know what you are thinking right about now: ahhhhh! What does it all mean? It's simple really. It means that if we have a square to leg a, and a square to leg b, then the sum of the area of these two squares is equal to the area of the square on the hypotenuse.
Since this is a square, then each side must measure a. The area of a square is equal to length times width.
So the area here is a square.
Similarly, this is a square, so each side must measure b. The area of this square is b square.
Finally, this square has sides of h, which makes the area h square.
And with this, we can see the Pythagorean equation. A square plus b square equals h square or in other words. Leg square plus leg square equals hypotenuse square.
Ok, we are ready to do some maths.
In this example, we must find the value of this. Let's start by identifying the sides of the right triangle. The hypotenuse can be located across the ninety degree angle.
In this example, x is the hypotenuse.
These two sides form the letter L, so these sides must be the legs.
Now I can set up my problem using the Pythagorean equation: leg square plus leg square equals hypotenuse square.
I know that six and eight are my legs. So, all right, six square plus eight square. X is my hypotenuse, so I finish by writing: equals x square.
Six square is six times six, eight square is eight times eight. I bring down my x square.
Six times six is thirty-six, eight times eight is sixty-four.
Now we are left with one hundred equals x square. Root squaring both sides of the equation and we have x equals ten.
Now let's try a word problem: A painter is on top of a thirty-five feet ladder that is leading against the house. The base of the ladder is twenty-one feet from the base of the house. If the painter were to fall, how far down would his fall be?
Drawing a picture is an excellent strategy to solving word problems. Here we have our painter on top of the ladder that is thirty-five feet long. The base of the ladder is twenty-one feet from the base of the house.
The problem is asking: if the painter were to fall, how far down would his fall be?
So we are looking for this side. Let's label this side x.
Notice this create a right triangle. So we can start this problem by using the Pythagorean equation.
Let's start by identifying the sides of the right triangle. We can locate the hypotenuse by looking across the ninety degree angle. In this case thirty-five is our hypotenuse. The remaining two sides must be my legs.
The Pythagorean equation says that leg square plus leg square equals hypotenuse square.
I know that twenty-one and x are my legs and that thirty-five is my hypotenuse. So, all right, twenty-one square plus x square equals thirty-five square. Twenty-one square is four hundred and forty-one, I bring down my x square and thirty-five square is one thousand two hundred and twenty-five.
Now we can subtract four hundred and forty-one from both sides of the equation to isolate the x variable. X square equals seven hundred and eighty-four.
Finally, we square root both sides of the equation and x equals twenty-eight.
So if the painter falls, he will fall twenty-eight feet down.
Earthquake drill!! Ahhhh!!!!
Let's do one final problem: John wants to put his tv for sale in craiglist. The only problem is that John forgot the inches of his tv. TVs are advertised by the inches of the diagonal. John knows the width of the tv is forty-five inches and the high is twenty-eight. How many inches should John advertised his tv as?
So, once more, we will draw a picture to illustrate the word problem. The width of the tv is forty-five inches, and the high is twenty-eight inches.
We are looking for inches of the diagonal.
Notice this picture creates a right triangle.
We can solve this problem by using the Pythagorean equation.
I'll start by looking for the hypotenuse. I look across the ninety degree angle, and there it is, x is the hypotenuse. These two sides must be my legs.
So all right, twenty-eight square plus forty-five square equals x square. Twenty-eight square is seven hundred and eighty-four. Forty-five square is two thousand twenty-five. Seven hundred and eighty-four plus two thousand twenty-five equals two thousand eight hundred and nine.
Now we can square root both sides of the equation and x equals fifty-three.
Now, John can advertise his fifty-three inches tv in craiglist.
Well, that is for a show. Thanks for watching. I'm Sergio Alvarez and I see you next time.
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