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We are on problem 12.
I had to take a pause because I had a hunger pang,
but I had to resist.
Because I do this thing called the Warrior Diet,
which is more of a way of life-- but anyway,back to the SAT.
But you shouldn't starve yourself
because you have to feed your brain for the SAT.
OK, let me draw the x- and y-axis.
Whenever I'm hungry, I just go outside and do some sprints
because that's the warrior way.
That's the y-axis.
But you shouldn't do that
because you're probably still growing.
I, unfortunately, have stabilized at 5'9".
All right, so let's see,
they drew something that looks like a parabola,
so I'm going to assume it is without better information.
So let me draw it.
My thing does not look like a parabola,
but I think you get the point.
And then they draw this rectangle-looking thing.
I'll draw it in another color.
Something like that. Let's see, let me label these points.
Points q, r, s, this is t.
And they say that this is-- oh, p, p not t.
And this is point negative 1, 0.
Obviously the y is 0 because we're on the x-axis
and that's negative 1 and that's 1.
OK,they tell us in the figure above
p, q, r, s is a rectangle
and points q and r lie on the graph of y equals ax squared.
And that's this thing, this yellow thing.
Where a is a constant.
If the perimeter of pqrs is 10,what is the value of a?
If we can figure out one ordered pair,
if we can figure out one x-y pair that lie on this curve,
we can figure out what the curve is
because we can substitute that y and x
into this equation and solve for a.
Well, do we know any points that are on that?
Well, both r and q are on it,
so if we can figure out the coordinates of r and q,
then we will be set. So what can we do?
Well, they told us a perimeter.
They told us the perimeter of this rectangle.
So what do we know about the rectangle already?
How long is this side right here?
Well, it goes 1 unit to the left on the x-axis
and it goes 1 to the right and it's flat.
So that has a width of 2, right?
It's a rectangle, so if that has a width of 2,
this also has a width of 2, right?
Let's call this height.
That's the height of the rectangle.
If that's the height, that's also the height.
So what's the perimeter?
The perimeter is 2 plus 2, which is 4, plus h plus h.
And that equals what? They told us that equals 10.
So we can add those h's together.
We get 4 plus 2h is equal to 10.
Or subtract 4 from both sides.
2h is equal to 6. Divide both sides by 2. h is equal to 3.
So we know that h equals 3.
So if h equals 3, what is the point r?
Well we know that its x-coordinate is 1 out,
so it's 1, and then what's the y-coordinate?
Well, this height right here is 3, right?
And we're starting at 0,so we go 3 up from 0.
So this is going to be 3 right here.
So the r's coordinate is 1, 3.
And r is on the yellow parabola,so now we are set.
We can take this and substitute x and y here.
We know y is equal to 3 when x is equal to 1.
So 1 squared, right? x squared.
So 3 is equal to a times 1 squared is 1.
So a is equal to 3 because that one doesn't matter, right?
a is equal to 3 so we just write down 3 inside there.
Next problem. Problem 13.
If a times b plus b is equal to a plus 2c,
what is the value of b
when a is equal to 2 and c is equal to 3?
So a real quick thing that we can do just to
maybe simplify things for us is we can factor out the b
out of this side.
What do I mean by factoring out a b?
Well, divide both of these terms by b
and you get b times a plus 1 is equal to a plus 2c.
All right, let me box this out so you don't get confused.
All I did is I undistribute the b,
right? You can multiply this back out
and you'd get ab is equal to b.
And then if you want to solve for b,
divide both sides by a plus 1
and you get b is equal to a plus 2c over a plus 1.
And then you substitute.
So a is 2, so it's 2 plus 2 times c,
c is 3, 2 times 3 over 2 plus 1,
which is equal to 2 plus 6 over 3, which is equal to what?
8/3.
Another way you could have done it--
and it might have been even faster,
actually--is just to substitute from the beginning.
So you say a is 2, so that would give us 2b plus b right?
I just, a is 2 is equal to 2 plus 2 times c, 2 plus 6.
So you get 2b plus b,so what's 2b plus b?
That's 3b. 3b is equal to 2 plus 6, 8.
Actually this would have been faster.
8 divided by 3.
So those are two ways you could have done it,
and hopefully you'll do them one of the two ways.
Hopefully you'll do it the faster way.
Next problem. 14.
Bunch of lines for me to draw.
I do so much for you guys.
The only thing I get in return is the satisfaction
of you guys getting a perfect score on the SAT.
I'm going to take these videos down before I have kids,
though, so they don't have competition.
Let's see, OK. I'm kidding.
The competition will only make them stronger.
I don't have kids yet, by the way, but one day I will.
And they'll have to compete against you guys,
who I've given this edge to.
That's how selfless I am.
And anyway, back to the problem.
So we have this-- I'm only partially serious.
OK, this is l, this is a, b, c.
This is angle y, y degrees.
This out here is angle x, x degrees.
In the figure above, m is parallel to n, so this is m,
this is n, they are parallel. Fair enough?
And l bisects angle abc.
So what does bisect mean?
That means that it splits the angle perfectly,
so that tells us that angle is equivalent to this angle.
OK?
If y is between 45 and 55, what is one possible value for x?
So let's just say that y is 50 and see what we can do here.
So if y is 50, right?
That satisfies, this is between 45 and 55.
What is this larger angle here
that I'm going to do in magenta
Well, it's a corresponding angle, right?
Because it's a parallel line,k is a transversal,
so this magenta angle corresponds to y.
So it will also be equal to 50.
And we know l bisects this angle,
so both of these smaller angles are going to be 25 and 25.
And now l is a transversal as well.
So if l is 25, this angle right here is 25.
Let me do that in another color.
No, that color was too similar.
I'll do it in red.
If this right here is 25, what angle is this going to be?
Well once again, corresponding angles of a transversal,
so this is also going to be 25.
And now what could x be?
Well, x is opposite to this angle,
so it's also going to be 25 degrees.
So this is just the angle game on steroids.
And you really didn't have to do the range,
you just pick a number. y is 50,
corresponding angle is also 50.
That corresponding angle is bisected,
so each of these smaller angles is 25.
And then l is a transversal again, so if this is 25,
this is 25, and x is opposite.
So it's also 25.
I'll see you in the next video.