CA Geometry: Area, Pythagorean Theorem 26-30, area, circumference, pythagorean theorem
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.
- 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.
- 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
- 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
- So 22 pi times 8 is 176 pi.
- And that's choice C.
- See you in the next video.
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