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## High school geometry

### Course: High school geometry>Unit 6

Lesson 3: Problem solving with distance on the coordinate plane

# Area of trapezoid on the coordinate plane

Sal finds the area of a trapezoid using the distance formula and the trapezoid area formula.

## Video transcript

- [Voiceover] So we have a trapezoid here on the coordinate plane and what we want to do is find the area of this trapezoid, just given this diagram. And, like always, pause this video and see if you can figure it out. Well, we know how to figure out the area of a trapezoid. We have videos where we derive this formula. But, the area of a trapezoid, just put simply, is equal to the average of the lengths of the bases, we could say base one plus base two times the height. And, so what are our bases here? What are, and what is going to be our height over here? Well, we could call base one, we could call that segment CL, so it would be the length of segment CL, right over here. I'll do that in magenta. That is going to be base one. Base two, base two, that could, we'll do that in a different color. Base two would be the length of segment OW, or b two would be the length of segment OW, right over there. And then, our height, our height h, well, that would just be an altitude and they did one in a dotted line, here. Notice it intersects the, base one, I guess you could say segment CL at a right angle, here. And so, this would be, this would be the height. So, if we know the lengths of each of these, if we know each of these values, which are the lengths of these segments, then we can evaluate the area of this actual trapezoid. And, once again, if this is completely unfamiliar to you or of you're curious, we have multiple videos talking about the proofs or how we came up with this formula. You an even break out, break down a trapezoid into two triangles and a rectangle, which is one way to think about it. Anyway, let's see how we could figure this out. So, the first one is, what is b one going to be? B one is the length of segment CL and you could say, "We'll look, we know what these, "the coordinates of these points are." You could say, "Let's use the distance formula," and you could say, "Well, the distance formula is just "an application of the Pythagorean Theorem." So, this is going to be the square root of our change in x squared, so our change in x is going to be this, right over here. And, notice, we're going from x equals negative four to x equals eight, as w go from C to L. So, our change in x is equal to eight minus negative four which is equal to 12. And, our change in in y, our change in y, we are going from y equals negative one to y equals five. So, we could say our change in y equal to five minus negative one which, of course, is equal to six. And, you see that: one, two, three, four, five, six. And, the segment that we care about, its length that we care about, that's just the hypotenuse of this right triangle that has one side 12 and one side six. So, the length of that hypotenuse, from the Pythagorean Theorem, and, as I mentioned, the distance formula is just an application of the Pythagorean Theorem, this is going to be our change in x, squared, 12 squared, plus our change in y, squared, so plus six squared, and this is going to be equal to 144 plus 36. So, the square root of 144 plus 36 is one hundred, one hundred and 80 which is equal to, let's see, 180 is 36 times five. So, that is six square roots, six square roots of five. Oh, let me not skip some steps. So, this is the square root of 36 times five which is equal to, square root of 36 is 6, so six square roots of five. Now, let's figure out b two. So, b two, once again, change in x squared plus the square root of change in x squared plus change in y squared. Well, let's see, if we're going from, we could set up a right triangle, if you like, like this, to figure those things out. So, our change in x, we're going from x is at negative two, x is going from negative two to positive four. So, our change in x is six. Our change in y, we are going from, we are going from y equals, y equals five to y equals eight. So, our change in y is equal to three. So, just applying the Pythagorean Theorem to find the length of the hypotenuse, here, is going to be the square root of change in x squared, six squared, plus change in y squared, plus three squared which is going to be equal to, it's going to be equal to 36 plus nine, which is 45, so, square root of 45 which is equal to the square root of nine times five which is equal to three square roots of five. And so, we only have one left to figure out. We have to figure out h. We have to figure out the length of h. So, h is going to be equal to, and so, what is our, if we're going from W to N, our change in x is two. Change in x is equal to two. We're going from X equals four to x equals six. If you want to do that purely numerically, we would say, "Okay, our endpoint, our x value is six. "Our starting point, our x-value is four. "Six minus four is two." You see that visually, here. So, it's going to be the square root of two squared plus our, let me write that radical a little bit better. So, it's the square root of our change in x squared plus our change in y squared. Our change in y is negative four. Change in y is negative four but we're going to square it, so it's going to become a positive 16. So, this is going to be equal to the square root of four plus 16. Square root of 20, which is equal to the square root of four times five, which is equal to two times the square root of five. It's nice that the square root of five keeps popping up. And so, now we just substitute into our original expression. And so, our Area of our trapezoid is going to be one half times six square roots of five, six square roots of five, plus three square roots of five, plus three square roots of five, let me close that parentheses, times two square roots of five, times two square roots of five. And, let's see how we can simplify this. So, six square roots of five plus three square roots of five, that is nine square roots of five. Let's see, the one half times the two, those cancel out to just be one. And so, we're left with nine square roots of five times the square root of five. Well, the square root of five times the square root of five is just going to be five. So, this is equal to nine times five which is equal to 45 square units or units squared.