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CA Geometry: Area, pythagorean theorem

26-30, area, circumference, pythagorean theorem. Created by Sal Khan.

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Video transcript

All right. We're on problem 26. For the quadrilateral shown below, a quadrilateral has four sides, measure of angle A plus the measure of angle C is equal to what? And here, you should know that the sum of all the angles in a quadrilateral are equal to 360 degrees. And you might say, OK, I'll add that to my memory bank of things to memorize. Like the angles in a triangle are equal to 180. And I'll show you no, you don't have to memorize that. Because if you imagine any quadrilateral, let me draw a quadrilateral for you. And this is true of any polygon. So let's say this is some quadrilateral. You don't have to memorize that the sum of the angles is equal to 360. Although it might be useful for a quadrilateral. But I'll show you how to always prove it for any polygon. You just break it up into triangles. Then you only have to memorize one thing. If you break it up into triangles, this angle plus that angle plus that angle has to be equal to 180. And this angle plus that angle plus that angle have to be equal to 180. So the angles in the quadrilateral itself are this angle and this angle. And then this angle and this angle. Well this one is just the sum of those two, and this one's just the sum of those two. So if these three added up to 180. And these three added up to 180. This plus this plus this, plus this will add up to 360. And you can do that with an arbitrarily shaped polygon. Let's do five sides, let's do a pentagon. So one, two, three, four, five sides. Wow, how many angles are there in a pentagon. Just break it up into triangles. How many triangles can you fit in it? Let's see. One, two. Each of these triangles, their angles, they add up to 180. So if you want to know that, that, that, plus that, that, that, plus that, that, and that. That would just be 180 times 3, which is 540. And that also would be the angle measures of the polygon. Because these three angles add up to that angle. That's that. Those angles add up to that one. Those angles add up to that one, and those angles add up to that one. So now hopefully, if I gave you a 20 sided polygon, you can figure out how many times can I fit triangles into it. And you'll know how many angles there are. And the sum of all of them. But anyway, back to the quadrilateral. A quadrilateral, the sum of the angles are going to be 360 degrees. So, if we say, measure of angle A, plus measure of angle C, plus these two angles. Let me write it down. Plus 95 plus 32 is going to be equal to 360. So I'll just write A plus C, just a quick notation. Let's see, 95 plus 32 is 127. Plus 127 is equal to 360. A plus C is equal to 360 minus 127. And what is that? That is 233. Right, and that's the choice. Fair enough. Question 27. If ABCD is a parallelogram, that's the sides are parallel, what is the length of segment BD? So they want from here to here. And this is just another interesting thing, I'm not going to prove it right now, but this is a good thing to know, especially if you become a mathlete. Because it shows up in math competitions every now and then. If you have a parallelogram, the opposite sides are parallel, then their diagonals are actually bisecting each other. Which means that they split the other diagonal in two. So this diagonal splits this diagonal in two. So if this is 6, this is also going to be 6. And this diagonal splits BD in two. So if this is 5, then this is also 5. So BE is 5, ED is 5, then BD has to be 10. Choice A. Let me copy and paste 28 in here. A right circular cone has radius 5 inches and height 8 inches. Fair enough, they've drawn it for us. What is the lateral area of the cone? Good, they gave us a definition. Lateral area of a cone is equal to pi times r times l, where l is the slant height. So we know what r is, they give us r. r is 5. So we just have to figure out what the slant height is, this l. Well this looks like a Pythagorean theorem problem. This is a right angle, I know it's all weird because it's three dimensions. But this forms a right triangle. We're just kind of picking one slice of the cone that includes the pointy part. We say 5 squared, plus 8 squared is equal to l squared. This is a right angle, l is the hypotenuse. So we get 25 plus 64 is equal to l squared. So that's 89 is equal to l squared. And so, l is equal to the square root of 89. Unless, I've made a mistake someplace. Square root of 89. Oh good, I see a square root of 89 there already. So we probably are on the right track. So l is equal to the square root of 89. And they give us the formula for the lateral area of a cone as pi r l. So pi r l is equal to pi times r, the radius of the base, which is 5. Times this slant height, which is the square root of 89. This equals 5 pi times the square root of 89. Which is, I just peeked and saw, choice D. Whenever you see a number like 89, you begin to get worried. But it's good that that was one of the choices. Problem 29. OK. Let me copy and paste it. Clear this image. It's early on a Saturday morning, my wife is still sleeping. We're expecting our first child in a month. So I figure the sleep is good for her. Gives me more time to record math videos. OK, I don't know why I go onto those tangents. OK, figure ABCD is a kite. And it looks like a kite. What is the area of figure ABCD in square centimeters. Well everything they're giving us is in centimeters. So if we just stay in centimeters we won't have a problem. So what's the area of this? So we just figure out the area of each of these triangles. And what's the area of a triangle? The area of a triangle is equal to 1/2 base times height. So what's the area of this triangle? Well, actually, this is symmetric. If we know the area of this triangle, we know the area of this triangle. Because this is 6 and 8, this is 6 and 8. So the area of this one is 6 times 8 is 48. 48 times 1/2 is 24. This one is also going to be 24 by the same argument. So when you add them together, you get 48. Those two combined are going to be 48. Now, this triangle, 8 times 15 times 1/2. That's 4 times 15, which is equal to 60. And this is going to have the same area by the same argument. 60. We don't even have to multiply by 1/2, because we're going to multiply by 2 eventually. Or add it to each other again. So anyway, we have 60 plus 60 plus 24 plus 24, that's 120 plus 48, so 168. Choice C. Next problem. Problem 31. I like these problems. Now that we're out of the whole part that they were getting into congruencies and similars. And I thought they made a couple of mistakes on some of those. Anyway, if a cylindrical barrel measures 22 inches in diameter, how many inches will it roll in 8 revolutions along a smooth surface? So we could imagine a wheel. It's a tire of some kind. So let me draw a circle. So if we look at the cylindrical barrel from the side, because I think that's all we care about. That's its side. They say, a cylindrical barrel measures 22 inches in diameter. So this distance right here. That distance right there is 22. And what they say is, this thing is going to roll 8 revolutions on a smooth surface. It's going to go around 8 times. It's going to roll and move to the right. So how long will it roll? So if you think about it, it's going to cover its circumference 8 times. If this point is starting touching the ground, after it moves a circumference of distance, that point will be touching the ground again. An easy way to think about it is, as this thing moves to the right, as it rolls, when it moves 1 foot, 1 foot along of circumference will then be touching the ground. Or 1 cm, or 2 inches or whatever. Then 2 inches along its circumference will be touching the ground. So it's going to go 8 circumferences in 8 revolutions. So what's the circumference of this? Circumference is equal to pi times the diameter. The diameter they already gave us is 22. So the circumference is equal to 22 pi. So it's going to move 8 circumferences in 8 revolutions. So 22 pi times 8 is 176 pi. And that's choice C. See you in the next video.