Lesson 10: Applications of the Pythagorean Theorem
A carpet measures 7 feet long and has a diagonal measurement of square root of 74 feet. Find the width of the carpet. So let's draw ourselves a carpet here. So let's draw a carpet. It has a length of 7 feet, so let's say that that is 7 feet, right there. And it's going to be a rectangle of some kind. So let's say that we're looking down on the carpet like that. That's our carpet. And then it has a diagonal measurement of square root of 74 feet. So that means that this distance, right here-- draw it a little bit neater than that --this distance right here, the diagonal of the carpet, is the square root of 74 feet. And what they want to know is the width of the carpet. Find the width of the carpet. So let's say that this is the width of the carpet. That is w, right there. Now, you might already realize that what I have drawn here is a right triangle. Let me make sure you realize it. This is a 90 degree angle here. And since that is a triangle that has a 90 degree angle, it's a right triangle. The side opposite the right angle, or the 90 degrees, is a hypotenuse, or the longest side. It is the square root of 74. And the shorter sides are w and 7. And the Pythagorean Theorem tells us that the sum of the squares of the shorter side will be equal to the square of the hypotenuse, so the square of the longer side. So we get w squared, this side squared. plus 7 squared, this other side squared, is going to be equal to the hypotenuse squared, square root of 74 squared. And then we get w squared plus 49 is equal to the square root of 74 squared. Well, that's just going to be 74. It is equal to 74. We can subtract 49 from both sides of this equation. So we have just a w squared on the left-hand side. Subtract 49 from both sides. The left side-- these guys are going to cancel out, we're just going to be left with a w squared --is equal to-- What's 74 minus 49? 74 minus 49, well, we can do a little bit of regrouping or borrowing here, if we don't want to do it in our head. We can make this a 14. This becomes a 6. 14 minus 9 is 5. 6 minus 4 is 2. And we have w squared is equal to 25. So w is going to be equal to the square root of 25, the positive square root. So let's take the square root of both sides, the positive square root, and we will get w is equal to 5. Because we obviously we don't want it to be negative 5. That wouldn't be a realistic distance. So the width of the carpet is 5. And we're done.