We’re on problem 36 and it says what is the area in square units of the trapezoid shown below. So when you just look at this you’re like okay trapezoid do I know the formula for the area of a trapezoid and you get all confused and all and obviously trapezoid I can break it up and do a rectangle in a triangle if I were to draw a line right here. And if I know the dimension of each of those I know the dimensions, I know the area of each of them and then I know the area of the entire thing.
Let’s see, what’s this height right here or this width I should say. Well were going from zero to what? X is equal to eight here as to where straight down from X is equal to eight Y is equals to five so this dimension is eight and then when we go from X is equal to eight to X is equal to 12 how far is that? Well that’s going to be four; right so this is four and this is eight. Fair enough and then how high is this rectangle, we’re going from Y is going to zero and Y is going to five, so that’s five and of course this if five as well. So we’re done, we’re ready to figure out the area, the area of the rectangle part is eight times five, that’s 40. The area of the triangle is five times four times one half, if we didn’t put that one half we’d be figuring out the area of this rectangle right there.
So its five times four is 20 times one half is ten, so the area of both of this combined is ten plus 40 and 50. Thirty seven, the figure below the square with four congruent parallelograms inside, this looks interesting. What is the area and square unit of the shaded portion? So the shaded portion is the whole square minus the area of the parallelogram, so the whole square that’s easy that’s 12 and since we know that its square so the width also have to be square right? The area of the whole square is 144 and now we have to figure out the area, if you know the area of the parallelograms, the area of all the parallelograms because they are congruent. So lets see if we can figure out the area of one of the parallelograms.
So there is actually a formula of a parallelogram, its actually just the base times the height and they will actually give us that, let me show you that they give us that because it might not be obvious to you. So let me try to draw it, so it’s a one side, it goes straight like that and come down like that, good enough. Okay now if I just look at this parallelogram, they tell us that the height here is three and I know it’s the height because they told me it’s a 90 degree angle and they tell us that the base is five. And I’m telling you that the area of a parallelogram is just the base times the height is equal to 15 but you shouldn’t take my word for it. That should make intuitive sense to you and the way to think about it intuitively is imagine if we were to take this part of the parallelogram if you just cut it right here. And if we were to move it over here, if we were to cut that off and move it over here.
Then the parallelogram would look something like this. You’d have the part that we didn’t cut off right and then you move off the cut off part over here. And now the dimensions of this base would be five and then this height would be three and the area of this rectangle is 15 and there is no reason why the area of this should be any different than that. We just rearranged it this parts, so that’s why the area of a parallelogram is just the base times the height. So the area of each of this parallelograms are just 15 so the area of all of them combined is 15 times four which is 60, so 144 minus 60 is what? 84, that’s choice D.
Problem 38, what is the area in square meters of the trapezoid shown below. So to figure out the area we could break it up to these rectangles and triangles to figure out the area of this rectangle we need to know its height and actually we’ll need that to figure out the area of the triangle as well. So what’s this height right there? We know that this distance is going to be six, it’s a rectangle, that distance is going to be six, if that distance is six and both of these are five. Both of these triangles here are going to be congruent right because this length is equals to this length and we also make this angle as equal to that angle. But anyway, so what are these let me draw it another color. What are the lengths of these two green sides lets call it X all right.
Well we know that there are going to be three and now we can use that information to figure out this height right there because if we just draw this triangle right there. That’s three, that’s five, this is some unknown side A and you might even recognize we’re going to use a Pythagorean Theorem and this is a very typical type of right triangle. So you might already guess what A is but we’ll solve for it.
Problem 39, what is the area in square inches in the triangle below? Interesting, okay so this is an equal amount of triangle all the size are equal and so we can actually say that since these two triangles are symmetric, we can say that’s equal to that. So this side is going to give and this side is going to five, if this is five then that’s ten, what is this side right here? Lets call it X, Pythagorean Theorem, this is the hypoteneous, so this is X squared plus five squared plus 25 is going to be equal to the hypoteneous squared so equal to a 100. X squared is equal to a 100 minus 25 is 75, X is equals to the square root of 75, 75 is 25 times three right so that’s equal to the square root of 25 times three which is equal to the square root of 25 times square root of three which is equals to five root of three.
Next problem, problem 40, the perimeter of two squares is on a ration of four to nine, what is the ratio between the ratios of the two squares? Let me draw two squares, so let’s say that these are X right and the sides of this one are Y. So they’re saying the perimeters of the two squares are in a ratio of two to nine so the perimeter of the first square is four X right? So the perimeter of the first square is four X, the perimeter of the second square is four Y so that’s the ratio of the perimeter of the first square to the perimeter of the second square and then that is equal to four to nine. That is equal to four to nine and they say what is the ration between the areas of the two square? So they want us to figure out the first square is X squared and the ratio right base times height, so they want us to figure out what that is equal to right?
Well this is X squared over Y squared, this is the same thing as X over Y squared. So if we can figure out what X over Y is equal to we can just get square and we’ll get X squared over Y squared, let’s try to do that. So they gave us this, let’s see, if we divide, this is simplified right? Four over four, the reason X over Y is equal to four over nine right? X over nine is equal to four over nine so lets substitute that here, so X squared over Y squared is equal to X over Y squared which is equal to four nines squared which is equal to 16 over 81 or the ratio of the areas of the two squares is 16 to 81, choice D.
Maybe we can fit one more problem in there, actually no, I’m over ten minutes, I’ll stop right there. See you in the next video.
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