 In this video, we're going to use what we just learned about medians. So in this problem we have three variables that we have to solve for, x, y, and z. And we're going to use the different properties of a median that we just talked about. So I'm going to start with x, and x is the median. So I know that in order to get x, I have to add the bases together, 17 plus 53, and divide that by 2, or multiply by half. So if I add 17 and 53, I get 70, and 70 divided by 2 is 35, so x has to be 35. Next I'm going to go to y, and we know that because the median is the midpoint of the leg, then we know that y has to equal 11. And over on the other side to find z, because again, the median is at the midpoint of the leg, I know that these two pieces have to be congruent. And since it tells us that the whole thing is 26, then z has to be half of that, because it has to be congruent. So z would be 13. In this example, it's important to notice that this is an isosceles trapezoid. So if I want to find angle A, because it's isosceles, I know that the base angles are congruent. So A has to equal 78 degrees. To find angle B, we did a problem similar to this in one of the other videos. And what we're going to do is know that the trapezoid angles have to add up to 360 degrees, because it's a quadrilateral. So I'm going to add 78, oops, hello, 78 plus 78, and that's going to give us 156 degrees. And then I'm going to subtract that from 360, so 360 minus 156, and that's 204 degrees. And because I know that these two angles are congruent, I'm going to take 204 and divide it by 2, and that gives me 102 degrees. So B is 102 degrees. Then to find Y, I'm sorry, to find X, I am going to use the median formula. And that is that if you add the bases together, so 15 and X are the bases, if you add those together and divide by 2, you'll get the length of the median, which is 24. So what I'm going to do here is I'm going to write a little equation. X plus 15 divided by 2 equals 24. You can solve this a couple different ways. You can either multiply both sides by 2 to get rid of the fraction, or you can think of 24 as a fraction and cross multiply to solve for X, whichever one you'd like to do, you can. So either way, you get X plus 15 equals 48, and then when you subtract 15 from both sides, you get 33. So the value of X on the base would have to be 33. And then to find Y, this mark right here says that it's congruent to this, and so that means this is 15. And because the median is at the midpoint of the leg, then Y would also have to be 15. Okay, in this last example, we are going to have to find X and Y. It doesn't matter which one you start with. I will go ahead and find X first. And so because this is the median, I know that if I add the bases together and divide by 2, I will get the 2X minus 2 value. So I'm going to do 5X minus 21 plus 3X minus 31 divided by 2 has to equal 2X minus 2. So I'm going to combine like terms. So we get 8X minus 52 and divide by 2 equals 2X minus 2. Then I'm going to multiply both sides by 2 to get rid of this fraction. So it's going to be 8X minus 52. Now you just have to make sure that when you multiply this other side by 2, you actually distribute. So it's going to equal 4X minus 4. And I'm running out of room. So let me move my screen up here. And so now we're just going to solve for X. So I'm going to subtract 4X on both sides and add 52 on both sides. So we get 4X equals 48. And then when we divide by 4, we get X equals 12. Okay, the other variable we have to find here is Y. And so because I know that the median is at the midpoint of the leg, I know that Y squared has to be congruent to 14Y minus 48. So I'm going to set those equal. And once again, because we have both a Y and a Y squared, we're going to have to factor and solve this. So the first thing we're going to do is set it equal to 0. So I'm going to subtract 14Y and add 48 to get everything on one side. And so that it's equal to 0. And we're going to factor. So what two numbers multiply to 48 and add to negative 14? So I know it has to be two negative numbers. And if you guessed negative 6 and negative 8, you would be correct. Negative 6 times negative 8 is 48. And when you add them, you get negative 14. So now we get two answers, Y equals 6 and Y equals 8. And once again, we have to make sure that both of those answers make sense. So I'm going to go back over here and plug 6 and 8 in for Y. Well, either way, we get 36 and 64. So those numbers both make sense. I'm going to plug 6 in here for Y, 14 times 6 and subtract 48. And you get 64. I'm sorry. You get 36. And if you plug the 8 in for Y, 14 times 8 minus 48, you get 64. So this is a situation where both of these answers make sense. So you would actually give both of them as your answers.