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Video transcript

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 could 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 length of the basis we could say base one plus base two times the height and so what what our base is 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 to base 2 that couldn't let me do that in a different color base two would be the length of segment Oh W or B two would be the length of segment Oh W right over there and then our height our height H well that would just be an altitude and they they did want an a dotted line here notice that intersects bit enough for 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 if you're curious we have multiple videos talking about the proofs or how we came up with this formula you can even break out a trap break down a trapezoid into two triangles in a rectangle which is one way to think about it but anyway let's see how we could figure this out so the first one is what is b1 going to be b1 is the length of segment CL and you could say well look we 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 just going to be our 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 4 to x equals 8 as we go from C to L so our change in X is equal to 8 minus negative 4 which is equal to 12 and our change in Y our change in Y we're going from y equals negative 1 to y equals 5 so we could say our change in Y is equal to 5 minus negative 1 which of course is equal to 6 and you see that here 1 2 3 4 5 6 and the segment that we care about it's length that we care about that's just the that's just the hypotenuse of this right triangle that has one side 12 and one side 6 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 6 squared and this is going to be equal to 144 plus 36 so the square root of of 144 plus 36 is one hundred 180 which is equal to let's see 180 is 36 times five so that is 6 square roots 6 square roots of 5 all right let me not skip some steps so this is the square root of 36 times 5 which is equal to square root of 36 is 6 so 6 square roots of 5 now let's figure out B 2 so B 2 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 2 X is going from negative 2 to positive 4 so our change in X is 6 our change in Y we are going from we are going from y equals y equals 5 to y equals 8 so our change in Y is equal to 3 so just applying the Pythagorean theorem to find the length of the here it's 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 30 36 plus 36 plus 9 which is 45 so square root of 45 which is equal to the square root of 9 times 5 which is equal to 3 square roots of 5 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 WN our change in X is 2 change in X is equal to 2 we're going from x equals 4 to x equals 6 if you want to do that purely numerical you'll say Oh care endpoint are X values 6 our starting point on X values for 6 minus 4 is 2 you see that visually here so it's going to be the square root of 2 squared plus R 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 4 change in Y is negative 4 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 4 plus 16 square root of 20 which is equal to the square root of 4 times 5 which is equal to 2 times the square root of 5 it's nice that the square root of 5 keeps popping up and so now we just substitute into our original expression and so our area of our trapezoid is going to be 1/2 times 6 square roots of 5 6 square roots of 5 plus 3 square roots of 5 plus 3 square roots of 5 let me close that parentheses times 2 square roots of 5 times 2 square roots of 5 and let's see how we can simplify this so 6 square root of 5 plus 3 square roots of 5 that is 9 square roots of 5 let's see the 1/2 times the 2 those cancel out to just be 1 and so we're left with 9 square roots of 5 times the square root of 5 well square root of 5 times square five is just going to be five so this is equal to nine times five which is equal to forty five square units or units squared