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Current time:0:00Total duration:9:45

We're on problem 135. I just did some pilates, so I'm
slightly out of breath. If x is less than 0, is
y greater than 0? All right, statement number
1, x/y is less than 0. So this is actually
pretty useful. They're telling us that x
is a negative number. So we have a negative here, and
we're dividing it by some other number. They're telling us that
the resulting number is still negative. So we have a negative
divided by something is equal to a negative. So I think you learned when you
learned negative numbers, if y was negative-- well,
if y was 0 this would be undefined-- and if y is
negative, if this was a negative, then when you divide
a negative by a negative, you would have a positive. So y has to be a positive
number, and it can't be 0. So this actually tells us. Statement 1 alone tells us that
y is greater than 0 just based on the fact that
you have a negative. It's just telling you that
a negative divided by y is still negative. This whole thing is
still negative. So y has to be a positive. Statement number 2 tells us y
minus x is greater than 0. Let's see, add x
to both sides. That tells us that y
is greater than x. Well that's not enough. It says if x is 0, is
y greater than 0? Maybe x is equal to minus 10
and y is equal to minus 9. So in this case, y wouldn't
be greater than 0. Or maybe x is equal to minus 10
and y is equal to plus 9. Both of these would satisfy
this condition. So we don't know. This doesn't tell us any
information on whether y is greater than 0. So statement 1 alone us
sufficient, and statement 2 doesn't do much for us. Problem 135, no, I already
did 135, 136. They've drawn a circle there,
so I guess I'll draw a circle as well. So that's the circle. They have a center like that
and then it's like that. There's a right angle
right here. They call this O. They call this X. They call this Z. Then they actually connect
those two points. Then they call this, this point
right here, that's Y. Let's see what they're asking. What is the circumference
of the circle above with center O? So they just want to know
the circumference. So if we could figure out
a radius, then we know circumference is 2 pi
times the radius, and we'd be all set. So let's see what information
they give us. Statement number 1, the
perimeter of triangle OXZ is 20 plus 10 square roots of 2. Now this is interesting. What do we know about this? Well we know that it's going to
be an isosceles triangle. We know that if this length
right here is X, that this length is also going to be X,
because this is a circle. The radius is constant, and each
of these sides are the radius of the circle. So those are both
going to be x. What is this going to be? We just know from Pythagorean
theorem, x squared plus x squared is going to be equal
to that side squared. Let's call that c squared. So you get 2x squared is equal
to c squared or c-- I'm calling this side right here
c-- is equal to the square root of 2, x times the
square root of 2. Whatever x is. So the perimeter of this entire
triangle is going to be two x's, x plus x, so it's going
to be 2x plus x square roots of 2. Statement 1 told us that this
perimeter is equal to this right here. It's going to be equal to 20
plus 10 square roots of 2. Well I think you could do a
little pattern matching here and solve for x. This is actually, even though
it might not look completely like it, this is a
linear equation. You can solve for it. Immediately that should be, oh,
statement 1 is sufficient. But you can just do a little
pattern matching and say, well, this works when
x is equal to 10. So actually, we are able to
figure out what x is equal to. Then as I said before, x is
the radius of this circle. Then the circumference of the
circle, which is what the whole question is about, that's
2 pi times the radius. Circumference is 2 pi
times the radius. We just figured out
the radius is 10. So it equals 20 pi. So statement 1 is sufficient. What does statement 2 do? Statement 2, the length
of arc XYZ is 5 pi. XYZ is 5 pi. So they're telling us that
this length is 5 pi. Well do we know what proportion
that is of the entire circumference? Well sure, because they tell
us that this is 90 degrees right here. This is 90 degrees of 360
degrees of a circle. So this, the arc length right
here, is exactly going to be 1/4 of the circle. We know that because 90
degrees is 1/4 of 360. So 5 pi is going to be equal
to 1/4 times the circumference. Or that the circumference is
going to be equal to 20 pi. So each statement independently,
statement 1 or statement 2, independently, is
sufficient to answer this question, the circumference
of the circle. 137, what is the value
of a to the fourth minus b to the fourth? Immediately just even looking,
one thing I'm tempted to do is rewrite this as a
squared squared minus b squared squared. Because I just glanced at
statement 1 and they had squares there. So I'm going to do that. Let me do different letters. You know that x squared minus y
squared is equal to x plus y times x minus y. So here, x is a squared
and y is b squared. So this is going to be equal
to-- this is just the original, this is going to be
equal to a to the squared-- a squared plus b squared times
a squared minus b squared. Then we could simplified
this again. So this is going to be equal to
a squared plus b squared. We could simplify this as a
plus b times a minus b. I think that's going to be
helpful just glancing at the two statements that
they gave us. So statement 1 tells us that a
squared minus b squared is equal to 16. So that tells us that just
this is equal to 16. That is equal to 16. That doesn't help me much
because I don't know what a squared plus b squared
is going to be. It just tells me the difference between these two squares. I don't know whether they're
integers or anything. Statement 2 tells me that
a plus b is equal to 8. Once again, that doesn't help
me tremendously by itself. If I just know that a plus b
is equal to 8, and I don't know what a minus b
is, I don't know what all of this is. But when I use them together,
this is interesting. Because we know that a plus b
times a minus b is equal to a squared minus b squared. Well they just told us that
a plus b is equal to 8. So a plus b is equal to 8. So 8 times a minus b is going to
be equal to a squared minus b squared, which statement
one told us was 16. So that actually tells us that
a minus b is equal to 2. So now we have two equations
with two unknowns, and we could solve for a and b. If we can solve for a and b,
then we can definitely tell you what a to the fourth minus
b to the fourth is. Just to show you that,
let me do it. So let's see, a minus
b is equal to 2. Add these equations. 2a is equal to 10. a is equal to 5. Then if a is equal to
5, b is equal to 3. So our original statement,
a to the fourth is 5 to the fourth. 5 to the fourth minus
3 to the fourth. So whatever-- that's what? 625 minus 81, I think. It doesn't matter. You can figure it out. So both statements combined
are sufficient to answer this question. See you in the next video.