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All right.
So how do we say half of a quantity?
That would be 1/2 x.
And so let's take a look at all the different places where
we might encounter stuff like that.
So before we can do the 1/2, let's
just look at area of a parallelogram.
So let's review what shape a parallelogram is.
A parallelogram is a shape with two pairs of parallel segments.
That segment and that segment is parallel.
That segment and that segment are parallel.
Now I want to compute the area of that.
So we only know how to find area of rectangles.
So maybe we can cut this up.
And let's say the height of a parallelogram
is defined as the length of the perpendicular drop
from any vertex to the opposite base.
And so when you do that, this height
is going to play a role into how we
find the area of parallelogram.
Let's take a look how.
I can take that piece and cut it from the parallelogram
and move it here.
I can take that piece and move it here.
So that's my parallelogram.
But this is still a parallelogram.
So I'm just going to move this.
So again, this is my b.
And this height is my h.
And now I suddenly have a rectangle.
So area of a rectangle is that length times that length.
So that's b times h.
So area of a parallelogram is the base times height.
Base times height is area of parallelogram.
All right.
Let's take a look at area of a triangle.
If I want area of a triangle-- and remember
I can use parallelograms now-- this is what I'm going to do.
I'm going to take this shape and place it here.
And you can see that makes it a parallelogram.
So this part here is basically half of a parallelogram.
And you can see that this height, same as parallelogram
height, and it's also the same as the height of this triangle.
So the height remains the same no matter
where you are in the parallelogram.
That's because these two segments are parallel.
So the area of a triangle is going to be what?
1/2 base times height.
Let's look at trapezoid.
Remember, trapezoid has exactly one pair of parallel lines.
So area of this triangle is 1/2 base times height.
And so that's the base and that's
the height for this triangle.
All right.
So take a look at this triangle.
This triangle has the same height as this one
because if I move that height here, look,
do you see how it's the same?
So I can put these together.
And you can see how this triangle plus that triangle
makes this trapezoid up here.
And I already know how to find an area of a triangle.
So this is 1/2 a times h.
This is 1/2 b times h.
And then we add them together.
So area of a trapezoid is given by this formula.
So let's see.
How would you write 6 more than twice the quantity?
You would write 6 plus.
So more means plus.
Twice means 2x.
So that would be a combination of what we did before.
Let's say I have a situation where I have a bucket.
And there is water dripping.
All right?
So I have 6 liters of water that I've already
collected in the bucket.
And the bucket is filling up.
So you have 2 liters, another 2 liters in the next hour.
And then it just keeps filling at the rate
of 2 liters per hour.
And I want to know after so many hours, like say after t hours,
how much water is in here?
Since it's 2 liters for every t hours,
2t is the amount of water that's filled into this bucket.
So we would write that as what?
6 plus 2t liters.
If I want r% of a quantity, you would translate that into r%.
We saw percents is a fraction over 100, so r/100 times x.
So 30% of x would be 30/100 times x or 0.30x.
x is less than y, so that would be just less-than symbol.
So example would be 3 is less than 10.
So I would write it like this.
But what if I say x less than y?
Not x is less than, but x less than y?
That would mean subtract x from y.
So if I say, give me a number that is 2 less than 8.
OK, 2 less than 8?
So that would be 8, 7, 6.
So you're going to go backwards 2.
So 8 minus 2 are 6.
More formulas you might want to make
sure you know that are in the book
because there's lots of homework exercises in there for you.
So take a look.
In the book, it will look this, the summary
of all the different geometric formulas
that as you might want to look at.
We've already done all the way up to here.
The only thing we haven't done was
volume of a cylinder and volume of a cube and pyramid
and sphere.
But you can do that on your own.
This section has lots of homework.
Make sure you do all of it.