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# CA Geometry: Area, pythagorean theorem

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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.