- [Voiceover] In the figure above, PQRS is a rectangle.
So that, let us note that these are
right angles right over there.
The area of triangle RST is seven.
So RST is seven.
And PT is equal to 2/5 PS.
What is the area of triangle PQRS?
So let's think about what information they've given us
and what we need to figure out.
So they tell us the area of triangle RST is seven.
So what is the area of triangle RST,
given these lengths right over here?
Well, triangle RST, the area of any triangle is
1/2 times the base times the height.
So what is the base of RST?
What we could view it as the length of segment ST.
So the area is going to be equal to 1/2
times the length of segment ST,
for that (mumbles) times the height
which is the length of segment RS.
So times RS and that's equal to seven.
And that's equal to seven.
And if we wanted to simplify this a little bit,
we could multiply both sides by two.
And so, we could get the length of segment ST times
the length of segment RS is equal to 14.
Just multiply both sides by two.
Two times 1/2 is one.
Two times seven is 14.
So this first piece of information is
that the product of ST and RS,
ST and RS is 14.
Then they tell us that PT, I'll do PT in orange,
then they tell us that PT is
equal to 2/5
times PS.
Now let's just say, I'm gonna use pink for,
I'll just use this dark blue for PS.
Times PS.
So that's that length right over there, that's PS.
2/5 times PS.
So that's the second piece of information.
Right over here, I just rewrote it.
And now, what are we gonna figure out?
We need the area of PQRS of this entire rectangle.
Well, the area of rectangle is going to be
its base times its height.
So the area, what we need to figure out is
going to be PS, the length of segment PS.
So I can write, so we need to figure out PS
times the height which is RS.
Times RS.
This is what we need to figure out.
So going back to what we know, so here we know PT
in terms of PS or PS in terms of PT,
here we have an expression with ST, RS, and 14.
So somehow, if I could turn this ST part of this expression,
if I could turn this into something that deals with PS,
then I'm gonna be in pretty good shape.
Now let's just look at this.
How does ST relate to PS?
So let me write this over here.
So S, the length of segment ST
is equal to the length of segment PS.
It's equal to the length of segment PS.
Minus the length of segment PT.
So I could substitute this in for ST
but now it has something as in terms of both PS and PT.
But lucky for us, we can also substitute out
the PT for something in terms of PS.
That's what this does right over here.
So this, we know that PT, the second piece of information
they've given us, PT is equal to 2/5 PS.
So we can rewrite this whole thing and saying ST,
the length of segment ST is equal
to the length of segment PS minus,
minus 2/5 times
the length of segment PS.
Minus 2/5 times the length of segment PS.
And once again, how did I do that?
Well, we know that that is equivalent to that
from this piece of information,
from this piece of information here.
Now what's powerful about this is
now I have ST purely in terms of PS.
So if I replace ST with what I have in terms of PS,
it looks like I'm gonna have some type
of product of PS and RS or something
that I can solve for the product of PS and RS.
Before I even do that, let's simplify this a little bit.
If I have one PS minus 2/5 PS
or we could even say 5/5 PS minus 2/5 PS,
we get ST is equal to 3/5,
3/5 times PS.
And now we are ready to substitute.
So instead of having an ST here,
we could write 3/5 PS
3/5 PS
times the length of segment RS.
These are all the, you know the...
3/5 times the length of segment PS
times the length of segment RS is equal to 14.
And now we could just solve for this product.
What is PS times RS?
Well, let's just multiply both sides
times the reciprocal of 3/5.
Times 5 over three.
And we get, I think we deserve a drum roll now,
we get the product of PS
and RS being equal to,
let's see, what's 14 times five?
It's gonna be 50 plus 20.
It's going to be 70.
So this is going to be equal to 70 over three.
And we are done.