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This video is provided as supplementary material
for courses taught at Howard Community College, and in this video I'm going to
talk but two special right triangles.
So here's the first one. I've drawn a right triangle here.
I labeled the right angle, the 90 degree angle, and the other two angles
are 45 degree angles. So this is sometimes called
a 45-45-90-degree triangle, and
sometimes it's called an isosceles right triangle.
It's an isosceles triangle because I've got two angles that are congruent
and it's also a right triangle. Since it's isosceles,
I also know that these two legs, the sides next to the right angle,
are going to be congruent as well, they'll have the same length.
So let's say that the length
of one of the legs is x. That means I automatically know the length of the other
leg,
because they both have the same length, so that's x as well.
Let's find the length of the hypotenuse.
So we've got the Pythagorean theorem, and that tells us that
the hypotenuse, which is usually called C,
is equal to the square root of
A-squared plus B-squared,
where A and B are the two legs. So using the information we have here,
we can just say that C equals the square root
of x-squared plus x-squared,
since the two legs both have a length of x.
So that's going to be the same as the square root of
2x squared and I can simplify that
as just x times the square root of 2.
So I'll label this hypotenuse
as x times the square root of 2.
Now we can take the relationship
that we've got here between the length the leg
and length of the hypotenuse and us e this to solve specific problems.
So if you were told, for instance, that the leg
was 15 meters, then you would know that the hypotenuse,
instead of x times the square root of 2, would be
15 times the square root of 2 meters.
Whatever you're told the leg is, the hypotenuse is going to be that same
length
times the square root of 2.
Okay, so let's see what happens if you're given the hypotenuse
and you want to find the leg. So let's say we're told the hypotenuse
is 5 inches.
So 5 inches is going to be the same
as x times the square root of 2.
We want to find out what x is.
So to solve this equation, 5 equals
x times the square root of 2, I'll just divide both sides of the equation
by the square root of 2. On the right side of the equation,
the square roots will cancel and I'll just have x
equals 5 over the square root of 2.
Now if you're given this as a problem, you may be told not to leave a radical sign in the
denominator.
If that's the case, what we're going to do
is a process called rationalizing the denominator
and for this problem all it means is multiplying, the fraction that we have,
the fraction that we have, 5 over the square root of 2,
by the fraction the square root of 2 over the square root of 2,
which is just equal to 1. So to multiply these two fractions, I'll multiply across
and that means that I'll have for my numerator
5 times the square root of 2,
and the denominator will just be the square root of 2 times the square root of 2,
which equals 2.
Now let's go onto the other kind of right triangle that I want to talk about.
This right triangle,
besides having a 90-degree angle, has a 30-degree
and a 60-degree angle, so this is sometimes called a 30-60-90-degree
triangle. To understand this, let's realize
that this is just one half
of an equilateral triangle.
An equilateral triangle has 3 sides that are the same length
and 3 angles are congruent, they're all 60 degrees.
If we cut that equilateral triangle in half,
one of those 60 degree angles
gets cut in half and becomes a 30-degree angle.
Now if we had this
equilateral triangle and one of the sides
had a length of 2x,
then
I would know that all the sides had a length of 2x.
So the base has a length of 2x, and one-half the base,
the part that's part of the original triangle, the 30-60-90-degree triangle,
1/2 of the base would have a length of just x.
Now I'll get rid of that
extra half that I've drawn in and get back to our original triangle,
the 30-60-90 triangle. and what I know about a triangle like this
is if one of the legs has a length of x
then the hypotenuse has a length of 2x.
So let's find the length of the other leg.
One of the ways we use the Pythagorean theorem when we want to find the length of a
leg...
we'll label that leg as
side A, and that means that A
is going to equal the square root of
the hypotenuse squared, or C squared,
minus the other leg squared,
or B squared. So using the lengths that
we have here, this long leg
is going to equal the square root of the hypotenuse squared,
which is 2x,
that whole thing squared,
minus B squared. B is x, so
minus x-squared.
So I want to square 2x. So I'll have the square root of
4x squared minus x-squared.
And that equals the square root of 3 x squared,
and I can simplify that to x
times the square root of 3.
So that means if the short leg
has a length of x, the long leg has a length
of x times the square root of 3.
Now just as we saw with the isosceles right triangle,
if I know the length of
that shorter leg... so if I'm given a problem and I'm told the shorter leg
is 7 centimeters long, then I know that the longer leg
is going to be 7 times the square root of
3 centimeters long. Whatever the shorter leg is,
I just plug that number in before
the square root of 3, I multiply that number by the square root of 3.
And for the hypotenuse I just double the number.
Okay, so let's see what happens if
we know the length of the longer leg. So let's say that we know
that this long leg is
7 yards. That means the longer leg...
I'm going to make that equal to x times the square root of 3,
and I'll find out what x is, the length of the shorter leg.
So I'll take this equation, 7 equals x times the square root of 3,
I'll solve for x by dividing both sides
by the square root of 3. The square roots on the right will cancel
and x is going to equal
7 over the square root of 3. And once again I can rationalize this, I can get rid
of the
radical sign in the denominator
by multiplying by the square root of 3
over the square root of 3, and that's going to mean
that, multiplying across, I get
7 times the square root of 3 over the square root of three times the square root of 3,
which is just 3.
And it would also mean that I knew the length of the hypotenuse,
which would be twice as much as the length of that short leg,
or 14 times the square root of 3 over 3.
Okay, I hope this helps. Take care,
I'll see you next time.