Which of the triangles, triangle DEF, triangle GHI, and triangle
JKL, are similar to triangle ABC?
And we have this triangle ABC.
They've given us, or they've just marked one of the angles,
and then they've given the lengths of the two sides
on either side of that angle.
And you might remember from geometry class,
you can prove that two triangles are
similar if you can show they have corresponding congruent
angles.
So let's say we're trying to compare
this triangle to this triangle, triangle JKL to triangle ABC.
So if you can show that they have two corresponding angles
that are congruent to each other,
and that the sides on either side, the ratio
between the sides on either side of that angle are the same,
then you're going to be dealing with a similar triangle.
So, for example, if we say that this angle corresponds
to this angle, and they've already
marked it to say that they're congruent,
so that's a pretty good starting point.
And if we could show that the ratio between this side
and this side, side KL is the same as the ratio between BA
and KJ, then we can say that these two
are similar triangles.
Now you might be thinking, well, how
did I decide to compare BC to KL and BA to KJ?
Well, both of these are the two sides that essentially make up
this angle, and BC is the shorter of the two.
These are the two sides that make up this angle,
both of these are the, I guess you could say,
they're segments of the rays, there
are parts of the rays that make up this angle if you were
to continue forever, and the blue one is the shorter one,
so that's the best candidate.
KL is the best candidate to correspond to BC.
Well in order to do this, we're going
to figure out the lengths of these sides.
We're going to have to do it for all
of these triangles right over here.
And they don't give us the lengths,
but this thing is sitting on a coordinate plane.
Now you could use the distance formula,
but I like to just cut the chase.
The distance formula is based on the Pythagorean theorem,
and they've drawn a grid right over here for us,
so we could just set up some right triangles to figure out,
we could make each of these sides
essentially the hypotenuse of right triangles.
So to figure out the length of KJ,
we could set up this right triangle right over here.
This height is 3, this length right over here
we could count it, it's 1, 2, 3, 4, 5, 6, 7, 8, 9 units wide.
If we couldn't count like that, we
could say, look, we're going from x
equals negative 6 to x equals 3, 3 minus negative 6
is equal to 9.
Let me make that clear.
And that's essentially where the distance formula comes from,
3 minus negative 6.
You're finding the change, your change in x and you're
going to square it, and then you have your change in y,
you're going to square it, and that's
going to be equal to this hypotenuse squared.
So let's think about-- so this is
going to be equal to the square root.
I'm just using the Pythagorean theorem here.
So [? that line, ?] I'll just call it so K,
let me just write it out, so that you don't think
I'm doing some type of voodoo.
So 3 squared plus-- let me do it a little higher,
because I have a feeling I'm going
to use a lot of real estate-- 3 squared plus 9 squared
is equal to the length of KJ squared.
3 squared is 27, sorry, not 27, 3 squared is 9.
9 squared is 81, and then that's going
to be equal to the length of KJ squared.
Or, we get KJ squared is equal to 90,
which is the same thing as 9 times 10.
And so KJ is going to be-- it's the square root
of this, so the square root of that
is going to be the square root of 9 times 10
is equal to the square root of 9 times
the square root of 10, which is equal to 3 square roots of 10.
So it's actually the exact same length as BA.
Now let's do the same thing for KL.
For KL we could set up another right triangle right over here.
So Its height is 3 and its width is 3.
This is 3 and 3 right over here.
So we could write 3 squared plus 3 squared
is equal to the length of KL squared.
This right over here is 18 is equal to KL squared,
and if you take the square root of both sides
18 is, of course, 9 times 2.
So this right over here, this is the square root of 18
is equal to KL.
Square root of 18 is the same thing
as square root of 9 times 2, that's
just 3 square roots of 2.
This is equal to 3 square roots of 2.
So this is 3 square roots of 2.
This is 3 square roots of 10.
This angle is congruent to this angle right over here.
So not only are ABC and JKL, not only are they similar,
but these are actually congruent triangles.
These are congruent triangles right over here,
but all congruent triangles are going to be similar.
So this one this one is definitely similar.
JKL is similar to ABC.
Now let's figure out all of these lengths up here.
Well you have, let's see, this is 1
and 1, so you could say 1 squared plus 1 squared
is equal to EF squared, is equal to the length of segment EF
squared.
Or you could say EF is equal to-- Let's see, this is 2,
so this is going to be the square root of 2.
So actually I should have done that in that blue color
because this will be the shorter,
we want this to be the shorter side if we're comparing the two
sides on either side of that angle.
And now let's look at ED.
So once again, we can set up a right triangle
that is 1 wide and 3 tall.
And so we could say 1 squared plus 3
squared is equal to the length of ED squared.
1 squared plus 3 squared, this is going to be 10.
So we get ED is equal to the square root of 10.
Now is the ratio between BC and EF the same as the ratio
between BA and ED?
So let's write this down.
So square root of 3, or 3 square roots of 2
over square root of 2 is equal to 3.
So the ratio between these two blue sides between BC and EF
is 3.
The ratio between these two orange sides
is 3 square roots of 10 over the square root of 10,
which is also 3.
So the ratio between these two corresponding sides
are the same, the angle in between is congruent,
so these two triangles are also similar.
So DEF is also similar to ABC.
So we have one more to think about.
So HI is a candidate to be corresponding to BC.
What's its length?
Well, let's see, this is a right triangle
that's width 2, height 2, so we could write 2 squared plus 2
squared is equal to the length of HI squared.
This is 4, this is 4, so 8 is equal to the length
of HI squared.
And then we can take the square root of both sides.
So the square root of 8 is equal to the length of HI,
or we could say the length of HI is equal to,
let's see this is 4 times 2, so the square root
of 4 times the square root of 2 is 2 square roots of 2.
So this is 2 square roots of 2.
And then the candidate to be corresponding to AB is HG,
it's the longer of the two sides on either side of this angle.
So HG, we can set up a right triangle that looks like this.
It's 2 wide and 1, 2, 3, 4, 5, 6 high.
So we can write, and let's scroll over
to the right a little bit, we can
write that 2 squared plus 6 squared
is equal to the length of HG squared.
So let's see, this is 4 plus 36 is
equal to the length of HG squared.
Or we get that HG is equal to the square root of 40.
Square root of 40, let's see, how can we simplify that?
That's the same thing as the square root of 4 times 10,
which is 2 times the square root of 10.
So HG is equal to 2 times the square root of 10.
Now let's compare the ratios.
The ratio between BC and HI is going
to be 3 square roots of 2 over 2 square roots of 2, which
is 3/2.
And the ratio between BA-- let me
do that in that orange color-- which is 3 square roots of 10,
and HG, which is 2 square roots of 10, is also 3/2.
So all of these triangles, not only are they similar to ABC,
but they're actually all similar to each other.