WEBVTT

00:00:01.476 --> 00:00:04.146 A:middle
>> Welcome to the Cypress
College Math Review

00:00:04.466 --> 00:00:06.306 A:middle
on special right triangles.

00:00:06.636 --> 00:00:11.046 A:middle
In the first objective,
you will learn

00:00:11.046 --> 00:00:15.536 A:middle
to determine the side lengths
of 45, 45, 90 degrees triangles.

00:00:18.276 --> 00:00:23.086 A:middle
Given the right triangle ABC
here with measure angle A

00:00:23.086 --> 00:00:27.816 A:middle
and measure angle C being
45 degrees, then we know

00:00:27.816 --> 00:00:31.056 A:middle
that triangle ABC is
an isosceles triangle.

00:00:31.986 --> 00:00:36.656 A:middle
That means segment AB is
congruent to segment BC.

00:00:37.766 --> 00:00:43.066 A:middle
So, if AB is x units,
then BC is also x units,

00:00:43.326 --> 00:00:45.146 A:middle
where x is an arbitrary number.

00:00:45.146 --> 00:00:49.786 A:middle
And, the length of segment of
AC, which is the hypotenuse,

00:00:50.506 --> 00:00:52.836 A:middle
is x times square
root of 2 units.

00:00:53.476 --> 00:00:56.516 A:middle
We can verify this using
the Pythagorean theorem.

00:00:57.896 --> 00:01:01.266 A:middle
The Pythagorean theorem
states that the square

00:01:01.266 --> 00:01:04.996 A:middle
of leg 1 plus the
square of leg 2 is equal

00:01:04.996 --> 00:01:07.106 A:middle
to the square of the hypotenuse.

00:01:08.516 --> 00:01:13.246 A:middle
Therefore, we have x squared
plus x squared is equal

00:01:13.246 --> 00:01:17.146 A:middle
to the hypotenuse square,
which is AC squared.

00:01:18.656 --> 00:01:20.756 A:middle
X squared plus x
squared is 2x squared.

00:01:26.046 --> 00:01:27.696 A:middle
And, to get AC by itself,

00:01:27.736 --> 00:01:28.976 A:middle
we take the square
root of both sides.

00:01:35.666 --> 00:01:39.806 A:middle
So, we have AC is equal
to the square root

00:01:39.806 --> 00:01:42.966 A:middle
of 2 times the square
root of x squared is x.

00:01:48.106 --> 00:01:53.136 A:middle
So, we have AC is x
times square root of 2.

00:01:53.276 --> 00:01:57.606 A:middle
So, in general, for a 45,
45, 90 degrees triangle,

00:01:58.506 --> 00:02:00.396 A:middle
the two legs are congruent.

00:02:00.746 --> 00:02:04.886 A:middle
That means the length of leg 1
is equal to the length of leg 2

00:02:05.006 --> 00:02:08.646 A:middle
and the length of the
hypotenuse is the length

00:02:08.646 --> 00:02:12.176 A:middle
of the leg times
square root of 2.

00:02:12.406 --> 00:02:14.486 A:middle
The reason that the 45, 45,

00:02:14.536 --> 00:02:18.816 A:middle
90 degrees triangle is a
special type of triangle is

00:02:18.886 --> 00:02:22.196 A:middle
because knowing only one of the
three sides of the triangle,

00:02:22.416 --> 00:02:26.116 A:middle
we can determine the other two
as opposed to different types

00:02:26.116 --> 00:02:30.886 A:middle
of right triangles where we must
know the length of two sides

00:02:31.086 --> 00:02:33.686 A:middle
in order to find the
length of the third side.

00:02:34.866 --> 00:02:36.976 A:middle
It is helpful to
memorize these formulas.

00:02:44.046 --> 00:02:46.506 A:middle
We want to find the values
of y and h in this example.

00:02:47.796 --> 00:02:50.166 A:middle
Note that we cannot use
the Pythagorean theorem,

00:02:50.756 --> 00:02:53.176 A:middle
A squared plus B
squared equals C square

00:02:54.066 --> 00:02:58.116 A:middle
or y squared plus 7 squared
is equal to h squared.

00:02:58.716 --> 00:03:01.456 A:middle
The reason is because
we have two unknowns

00:03:01.726 --> 00:03:05.226 A:middle
and only one equation and
it's not possible to solve.

00:03:06.566 --> 00:03:09.846 A:middle
So, we will use the fact
that it is a 45, 45,

00:03:09.846 --> 00:03:12.206 A:middle
90 degrees triangle,
which is special.

00:03:12.636 --> 00:03:14.886 A:middle
This is an isosceles triangle

00:03:15.106 --> 00:03:16.976 A:middle
where the two legs
are congruent.

00:03:17.926 --> 00:03:20.976 A:middle
That means y is equal
to 7 units.

00:03:28.046 --> 00:03:30.646 A:middle
We see that h is the side
opposite of the right angle,

00:03:30.866 --> 00:03:32.396 A:middle
so it is the hypotenuse.

00:03:32.976 --> 00:03:37.286 A:middle
And, from the formula
you saw previously,

00:03:38.126 --> 00:03:44.326 A:middle
the length of the hypotenuse
is equal to the length

00:03:44.326 --> 00:03:46.406 A:middle
of the leg times
square root of 2.

00:03:47.756 --> 00:03:50.496 A:middle
Hypotenuse is h, the length

00:03:50.496 --> 00:03:53.496 A:middle
of the leg is 7 times
square root of 2.

00:03:54.836 --> 00:03:56.976 A:middle
And, this is just 7 rad 2 units.

00:04:06.576 --> 00:04:10.476 A:middle
We want to find the values of
c and v in this right triangle.

00:04:12.156 --> 00:04:15.246 A:middle
This angle is 45
degrees, which implies

00:04:15.246 --> 00:04:19.686 A:middle
that this angle is also 45
degrees, because the sum

00:04:19.686 --> 00:04:22.366 A:middle
of these two angles
must be 90 degrees.

00:04:24.126 --> 00:04:26.426 A:middle
Hence, this is a
special right triangle.

00:04:28.256 --> 00:04:33.526 A:middle
We know that the length of leg 1
is equal to the length of leg 2.

00:04:35.036 --> 00:04:40.076 A:middle
And, the length of the
hypotenuse is equal

00:04:40.076 --> 00:04:43.186 A:middle
to the length of a leg
times square root of 2.

00:04:45.006 --> 00:04:46.226 A:middle
Looking at the diagram,

00:04:46.226 --> 00:04:49.866 A:middle
we see that these two sides are
the legs and it doesn't matter

00:04:49.866 --> 00:04:54.626 A:middle
which one we call leg 1 or
leg 2, because they are equal.

00:04:55.706 --> 00:05:01.296 A:middle
Let's just call leg 1 C. Then,
the length of leg 2 is 3 rad 2,

00:05:01.996 --> 00:05:04.986 A:middle
therefore, C is rad 2 units.

00:05:06.826 --> 00:05:08.396 A:middle
This side is the hypotenuse

00:05:08.436 --> 00:05:10.546 A:middle
because it is opposite
the right angle.

00:05:11.226 --> 00:05:16.516 A:middle
So, we have v is equal
to the length of a leg,

00:05:16.516 --> 00:05:20.196 A:middle
which is 3 rad 2 times rad 2.

00:05:21.416 --> 00:05:23.936 A:middle
The product of rad
2 and rad 2 is 2.

00:05:29.056 --> 00:05:31.926 A:middle
You can do a little side work
here, that the square root

00:05:31.926 --> 00:05:35.246 A:middle
of 2 times the square root
of 2 is the square root

00:05:35.246 --> 00:05:37.636 A:middle
of 4, which is equal to 2.

00:05:38.986 --> 00:05:40.576 A:middle
As a short cut, the product

00:05:40.576 --> 00:05:44.976 A:middle
of any two same radicals is
the radicand, so in this case,

00:05:44.976 --> 00:05:49.166 A:middle
rad 2 times rad 2 is 2,
where 2 is the radicand.

00:05:49.786 --> 00:05:54.096 A:middle
Or, rad 5 times rad 5 is 5.

00:05:54.906 --> 00:06:02.726 A:middle
Rad x times rad x is equal to
x. So, back to the problem.

00:06:03.166 --> 00:06:07.886 A:middle
3 times 2 is 6, so v
is equal to 6 units.

00:06:15.086 --> 00:06:19.276 A:middle
Once again, we have a 45, 45,
90 degrees right triangle here.

00:06:19.626 --> 00:06:22.216 A:middle
Even though the measure of
this angle is not given,

00:06:22.456 --> 00:06:26.536 A:middle
but we do know that it is
45 degrees, because the sum

00:06:26.536 --> 00:06:29.046 A:middle
of these two angles
must be 90 degrees,

00:06:29.246 --> 00:06:31.206 A:middle
as we saw in the
previous example.

00:06:31.746 --> 00:06:34.846 A:middle
This side is the hypotenuse

00:06:35.136 --> 00:06:37.366 A:middle
because it is opposite
the right angle.

00:06:38.936 --> 00:06:45.216 A:middle
And, these two sides are the
legs and they are congruent.

00:06:45.886 --> 00:06:52.646 A:middle
We have the length of leg 1 is
equal to the length of leg 2

00:06:52.746 --> 00:06:59.046 A:middle
and the length of the hypotenuse
is equal to the length

00:06:59.046 --> 00:07:02.716 A:middle
of the leg times
square root of 2.

00:07:03.126 --> 00:07:06.306 A:middle
Now, we put in the variables
or values for the quantities

00:07:06.306 --> 00:07:07.616 A:middle
in these two equations.

00:07:08.586 --> 00:07:12.296 A:middle
Leg 1 is x or y,
it doesn't matter.

00:07:13.306 --> 00:07:17.526 A:middle
And, the other leg is
y. We have two unknowns

00:07:17.526 --> 00:07:21.396 A:middle
and one equation here, so it's
not going to get us anywhere.

00:07:22.696 --> 00:07:29.246 A:middle
The hypotenuse is 8 and the leg
is x or you can choose to put it

00:07:29.246 --> 00:07:32.606 A:middle
as y if you want to
times square root of 2.

00:07:33.496 --> 00:07:38.206 A:middle
In this second equation, we have
one unknown, so we can solve

00:07:38.206 --> 00:07:40.976 A:middle
for x by dividing the equation
by the square root of 2.

00:07:46.086 --> 00:07:51.286 A:middle
We get x is equal to 8 divided
by the square root of 2.

00:07:53.236 --> 00:07:55.366 A:middle
We need to rationalize
this number,

00:07:55.366 --> 00:07:57.776 A:middle
because we cannot have
radical in the denominator.

00:07:59.146 --> 00:08:02.386 A:middle
And, to rationalize it, we
will multiply the fraction

00:08:02.386 --> 00:08:05.256 A:middle
by square root of 2
over square root of 2.

00:08:06.716 --> 00:08:08.756 A:middle
Note that this quantity
is just 1.

00:08:08.756 --> 00:08:11.986 A:middle
It doesn't change the value
of the original fraction.

00:08:13.246 --> 00:08:18.256 A:middle
So, we have 8 times square root
of 2 is 8 square root of 2,

00:08:18.536 --> 00:08:20.926 A:middle
rad 2 times rad 2 is 2.

00:08:22.376 --> 00:08:26.136 A:middle
We can then reduce
8 divided by 2 is 4.

00:08:27.716 --> 00:08:30.916 A:middle
So, x is equal to 4 rad 2 units.

00:08:32.756 --> 00:08:34.876 A:middle
Now, we go back to
the first equation.

00:08:35.836 --> 00:08:40.916 A:middle
We already found the value
of x, so we just plug it in.

00:08:41.456 --> 00:08:44.976 A:middle
That means, y is equal
to 4 rad 2 as well.

00:08:51.396 --> 00:08:52.946 A:middle
And, this completes the problem.

00:08:57.166 --> 00:08:59.976 A:middle
Pause the video and
try these problems.

00:09:08.046 --> 00:09:10.086 A:middle
In objective 2, you
are going to learn

00:09:10.086 --> 00:09:12.806 A:middle
to determine the side
lengths of a second type

00:09:12.806 --> 00:09:16.026 A:middle
of special right
triangle, which are the 30,

00:09:16.026 --> 00:09:18.266 A:middle
60, 90 degrees triangles.

00:09:18.586 --> 00:09:24.076 A:middle
Let triangle ABC be a
right triangle with measure

00:09:24.136 --> 00:09:27.856 A:middle
of angle A equals 30
degrees and the measure

00:09:27.856 --> 00:09:30.226 A:middle
of angle C equals 60 degrees.

00:09:30.816 --> 00:09:34.496 A:middle
Then, of course, the measure
of angle B is 90 degrees,

00:09:34.596 --> 00:09:36.126 A:middle
because it is a right angle.

00:09:38.036 --> 00:09:40.306 A:middle
You have learned
that in a triangle,

00:09:40.536 --> 00:09:44.286 A:middle
the side opposite the smallest
angle is the shortest side

00:09:44.926 --> 00:09:48.106 A:middle
and the side opposite
the largest angle is the

00:09:48.106 --> 00:09:48.956 A:middle
longest side.

00:09:50.626 --> 00:09:54.006 A:middle
Therefore, BC would
be the shortest side

00:09:54.186 --> 00:09:57.456 A:middle
because it is opposite
the 30 degrees angle.

00:09:59.536 --> 00:10:02.506 A:middle
And, since it is a leg
of a right triangle,

00:10:02.686 --> 00:10:04.496 A:middle
so we call it the short leg.

00:10:04.616 --> 00:10:09.306 A:middle
And, the other leg is
called the long leg.

00:10:10.366 --> 00:10:13.696 A:middle
Because it is the side
opposite the 60 degrees angle,

00:10:14.016 --> 00:10:16.756 A:middle
which is larger than
the 30 degrees angle,

00:10:18.596 --> 00:10:20.906 A:middle
and AC is the hypotenuse.

00:10:22.116 --> 00:10:24.436 A:middle
It is the longest side
of a right triangle.

00:10:26.286 --> 00:10:29.806 A:middle
Now, suppose that the length
of a short leg is some x units,

00:10:30.416 --> 00:10:33.556 A:middle
then the length of the
long leg is the length

00:10:33.556 --> 00:10:38.596 A:middle
of the short leg times square
root of 3 or it is x rad 3.

00:10:40.076 --> 00:10:40.616 A:middle
And, the length

00:10:40.616 --> 00:10:43.846 A:middle
of the hypotenuse is
always 2 times the length

00:10:43.846 --> 00:10:47.396 A:middle
of the short leg or in
this case, it is 2x.

00:10:48.936 --> 00:10:51.856 A:middle
This is the special
relationship of a 30,

00:10:51.856 --> 00:10:53.876 A:middle
60, 90 degrees triangle.

00:10:54.676 --> 00:10:58.646 A:middle
We can verify this relationship
using the Pythagorean theorem.

00:10:59.196 --> 00:11:13.016 A:middle
Short leg squared plus long
leg squared should equal

00:11:13.016 --> 00:11:18.066 A:middle
to hypotenuse squared,
which is commonly known

00:11:18.066 --> 00:11:20.876 A:middle
as a squared plus b
squared is c squared.

00:11:22.716 --> 00:11:24.406 A:middle
The length of the
short leg is x,

00:11:26.166 --> 00:11:32.736 A:middle
the length of the long leg is
x rad 3, and we want to know

00:11:32.736 --> 00:11:39.396 A:middle
if this is really the hypotenuse
squared or 2x squared.

00:11:41.226 --> 00:11:42.816 A:middle
Bring this x squared down.

00:11:44.036 --> 00:11:46.006 A:middle
The square of x is x squared.

00:11:46.776 --> 00:11:49.076 A:middle
The square of rad 3 is 3.

00:11:51.486 --> 00:11:55.216 A:middle
The square of 2x is 4x squared.

00:11:56.826 --> 00:11:59.926 A:middle
X squared times 3 is 3x squared.

00:12:00.356 --> 00:12:05.816 A:middle
And, indeed, x squared plus
3x squared is 4x squared,

00:12:06.096 --> 00:12:08.176 A:middle
which is equal to
the right-hand side.

00:12:09.786 --> 00:12:13.706 A:middle
We've verified the special
relationship among the sides

00:12:13.706 --> 00:12:16.966 A:middle
of the 30, 60, 90
degrees right triangle.

00:12:19.036 --> 00:12:21.526 A:middle
You should memorize
these relationships.

00:12:22.266 --> 00:12:25.966 A:middle
And, we will learn to apply
them in the following examples.

00:12:30.076 --> 00:12:34.106 A:middle
In this example, we want to find
the values of y and h in a 30,

00:12:34.106 --> 00:12:36.056 A:middle
60, 90 degrees triangle.

00:12:36.056 --> 00:12:39.966 A:middle
The first thing we need to
do is to identify the sides.

00:12:40.886 --> 00:12:44.256 A:middle
This is the short leg,
because it is opposite the 30

00:12:44.256 --> 00:12:45.196 A:middle
degrees angle.

00:12:46.046 --> 00:12:49.126 A:middle
This is the long leg,
because it is opposite the 60

00:12:49.126 --> 00:12:50.046 A:middle
degrees angle.

00:12:50.556 --> 00:12:52.356 A:middle
And, this is the hypotenuse,

00:12:52.446 --> 00:12:54.436 A:middle
because it is opposite
the right angle.

00:12:55.046 --> 00:12:59.086 A:middle
Now, we'll use the relationships
among the sides in a 30,

00:12:59.086 --> 00:13:01.906 A:middle
60 degrees triangles
that we saw earlier,

00:13:02.476 --> 00:13:11.776 A:middle
that is the long leg is equal to
the short leg times square root

00:13:11.776 --> 00:13:15.886 A:middle
of 3 and the hypotenuse
is 2 times the short leg.

00:13:17.806 --> 00:13:19.566 A:middle
Now, we plug in the variable.

00:13:20.136 --> 00:13:22.776 A:middle
Long leg is y, short leg is 7.

00:13:28.046 --> 00:13:32.916 A:middle
Hypotenuse is h and
short leg is 7.

00:13:33.536 --> 00:13:39.776 A:middle
Y is then equal to 7
rad 3 units and h is 14.

00:13:48.046 --> 00:13:50.136 A:middle
This is similar to
the previous example.

00:13:50.656 --> 00:13:51.766 A:middle
So, the first thing we need

00:13:51.766 --> 00:13:55.476 A:middle
to do is once again identify
the sides in the triangle.

00:13:56.046 --> 00:13:59.276 A:middle
This is the short leg,
because it is opposite the 30

00:13:59.276 --> 00:14:00.246 A:middle
degrees angle.

00:14:00.646 --> 00:14:03.506 A:middle
This is the long leg,
because it is opposite the 60

00:14:03.506 --> 00:14:04.416 A:middle
degrees angle.

00:14:04.836 --> 00:14:06.246 A:middle
And, this is the hypotenuse.

00:14:06.946 --> 00:14:09.766 A:middle
Now, we write down the equations

00:14:09.766 --> 00:14:11.366 A:middle
of the relationships
of the sides.

00:14:11.656 --> 00:14:13.726 A:middle
And, I do recommend that
you write these down,

00:14:13.806 --> 00:14:16.586 A:middle
because it will help you
in the calculation process.

00:14:18.296 --> 00:14:24.666 A:middle
The long leg is y and the short
leg is x. The hypotenuse is 8.

00:14:25.026 --> 00:14:29.406 A:middle
Now, between these
two equations,

00:14:29.736 --> 00:14:33.056 A:middle
we see that we can solve for
x in the second equation.

00:14:34.686 --> 00:14:37.976 A:middle
So, we divide both sides by
2 and get x is equal to 4.

00:14:42.206 --> 00:14:45.196 A:middle
Now, substitute 4 into
x. In this equation,

00:14:45.196 --> 00:14:48.976 A:middle
we get y is equal
to 4 rad 3 units.

00:14:56.126 --> 00:14:59.946 A:middle
Once again, we begin by
first identifying the sides.

00:15:00.576 --> 00:15:06.266 A:middle
This is the short leg,
this is the long leg,

00:15:06.266 --> 00:15:09.476 A:middle
and this is the hypotenuse.

00:15:14.266 --> 00:15:16.256 A:middle
Then, we have this relationship.

00:15:17.936 --> 00:15:21.146 A:middle
Now, we substitute the
variables and number in.

00:15:21.756 --> 00:15:30.326 A:middle
The long leg is 5, the short
leg is x. The hypotenuse is y

00:15:31.036 --> 00:15:36.316 A:middle
and the short leg is
x. Then, we will solve

00:15:36.316 --> 00:15:38.156 A:middle
for x in this equation.

00:15:39.626 --> 00:15:42.936 A:middle
To do so, we divide both
sides by the square root of 3.

00:15:56.496 --> 00:16:00.886 A:middle
We get x is equal to 5
over the square root of 3.

00:16:02.196 --> 00:16:03.816 A:middle
We need to rationalize this

00:16:03.866 --> 00:16:06.356 A:middle
because we cannot have
radical in the denominator.

00:16:07.256 --> 00:16:09.396 A:middle
To do so, we will
multiply top and bottom

00:16:09.396 --> 00:16:09.976 A:middle
by the square root of 3.

00:16:14.046 --> 00:16:17.516 A:middle
Note that square root of 3
over the square root of 3 is 1.

00:16:17.736 --> 00:16:21.376 A:middle
So, the product of 5 over
rad 3 times 1 is itself.

00:16:23.386 --> 00:16:24.816 A:middle
Now, we multiply across.

00:16:25.086 --> 00:16:28.166 A:middle
5 times the square
root 3 is 5 rad 3.

00:16:28.556 --> 00:16:31.596 A:middle
In the bottom, rad
3 times rad 3 is 3.

00:16:32.866 --> 00:16:34.666 A:middle
This cannot be simplified
anymore,

00:16:34.856 --> 00:16:40.206 A:middle
so x is 5 rad 3 over 3 units.

00:16:40.206 --> 00:16:43.966 A:middle
We substitute this value of x
into this equation to find y.

00:16:51.066 --> 00:16:53.086 A:middle
We multiply the numerators
together,

00:16:53.846 --> 00:16:56.156 A:middle
then we get 2 times 5 is 10.

00:16:57.626 --> 00:16:59.396 A:middle
And, in the bottom,
we just get 3.

00:16:59.556 --> 00:17:03.206 A:middle
This cannot be simplified
further, therefore,

00:17:03.206 --> 00:17:05.976 A:middle
y is 10 rad 3 over 3 units.

00:17:14.046 --> 00:17:17.446 A:middle
In this example, the
hypotenuse is given to us

00:17:18.966 --> 00:17:22.956 A:middle
and we're supposed to find
the lengths of the two legs.

00:17:24.386 --> 00:17:27.616 A:middle
X is the short leg, because
it is opposite the 30

00:17:27.616 --> 00:17:28.536 A:middle
degrees angle.

00:17:29.386 --> 00:17:34.746 A:middle
Y is the long leg, because it's
opposite the 60 degrees angle.

00:17:37.336 --> 00:17:39.446 A:middle
Then, we'll use this
relationship once again.

00:17:42.196 --> 00:17:46.416 A:middle
The long leg is y,
the short leg is x,

00:17:49.306 --> 00:17:52.286 A:middle
they hypotenuse is
6 square root of 5,

00:17:54.236 --> 00:17:59.936 A:middle
and the short leg is x. Now, we
can solve for x in this equation

00:17:59.936 --> 00:18:00.976 A:middle
by dividing both sides by 2.

00:18:15.046 --> 00:18:20.716 A:middle
Now, we can simplify this,
6 divided by 2 we get 3.

00:18:20.956 --> 00:18:23.976 A:middle
So, x is equal to
3 square root of 5.

00:18:29.336 --> 00:18:31.926 A:middle
Now, we substitute
this value into x

00:18:31.926 --> 00:18:32.976 A:middle
in this equation to get y.

00:18:44.476 --> 00:18:48.266 A:middle
We can multiply the two radicals
together and get 3 square root

00:18:48.266 --> 00:18:52.786 A:middle
of 5 times 3 or it is
3 square root of 15.

00:18:53.366 --> 00:18:54.976 A:middle
And, this is the value of y.

00:19:02.366 --> 00:19:04.976 A:middle
Pause the video and
try these problems.