>> Welcome to the Cypress College Math Review
on special right triangles.
In the first objective, you will learn
to determine the side lengths of 45, 45, 90 degrees triangles.
Given the right triangle ABC here with measure angle A
and measure angle C being 45 degrees, then we know
that triangle ABC is an isosceles triangle.
That means segment AB is congruent to segment BC.
So, if AB is x units, then BC is also x units,
where x is an arbitrary number.
And, the length of segment of AC, which is the hypotenuse,
is x times square root of 2 units.
We can verify this using the Pythagorean theorem.
The Pythagorean theorem states that the square
of leg 1 plus the square of leg 2 is equal
to the square of the hypotenuse.
Therefore, we have x squared plus x squared is equal
to the hypotenuse square, which is AC squared.
X squared plus x squared is 2x squared.
And, to get AC by itself,
we take the square root of both sides.
So, we have AC is equal to the square root
of 2 times the square root of x squared is x.
So, we have AC is x times square root of 2.
So, in general, for a 45, 45, 90 degrees triangle,
the two legs are congruent.
That means the length of leg 1 is equal to the length of leg 2
and the length of the hypotenuse is the length
of the leg times square root of 2.
The reason that the 45, 45,
90 degrees triangle is a special type of triangle is
because knowing only one of the three sides of the triangle,
we can determine the other two as opposed to different types
of right triangles where we must know the length of two sides
in order to find the length of the third side.
It is helpful to memorize these formulas.
We want to find the values of y and h in this example.
Note that we cannot use the Pythagorean theorem,
A squared plus B squared equals C square
or y squared plus 7 squared is equal to h squared.
The reason is because we have two unknowns
and only one equation and it's not possible to solve.
So, we will use the fact that it is a 45, 45,
90 degrees triangle, which is special.
This is an isosceles triangle
where the two legs are congruent.
That means y is equal to 7 units.
We see that h is the side opposite of the right angle,
so it is the hypotenuse.
And, from the formula you saw previously,
the length of the hypotenuse is equal to the length
of the leg times square root of 2.
Hypotenuse is h, the length
of the leg is 7 times square root of 2.
And, this is just 7 rad 2 units.
We want to find the values of c and v in this right triangle.
This angle is 45 degrees, which implies
that this angle is also 45 degrees, because the sum
of these two angles must be 90 degrees.
Hence, this is a special right triangle.
We know that the length of leg 1 is equal to the length of leg 2.
And, the length of the hypotenuse is equal
to the length of a leg times square root of 2.
Looking at the diagram,
we see that these two sides are the legs and it doesn't matter
which one we call leg 1 or leg 2, because they are equal.
Let's just call leg 1 C. Then, the length of leg 2 is 3 rad 2,
therefore, C is rad 2 units.
This side is the hypotenuse
because it is opposite the right angle.
So, we have v is equal to the length of a leg,
which is 3 rad 2 times rad 2.
The product of rad 2 and rad 2 is 2.
You can do a little side work here, that the square root
of 2 times the square root of 2 is the square root
of 4, which is equal to 2.
As a short cut, the product
of any two same radicals is the radicand, so in this case,
rad 2 times rad 2 is 2, where 2 is the radicand.
Or, rad 5 times rad 5 is 5.
Rad x times rad x is equal to x. So, back to the problem.
3 times 2 is 6, so v is equal to 6 units.
Once again, we have a 45, 45, 90 degrees right triangle here.
Even though the measure of this angle is not given,
but we do know that it is 45 degrees, because the sum
of these two angles must be 90 degrees,
as we saw in the previous example.
This side is the hypotenuse
because it is opposite the right angle.
And, these two sides are the legs and they are congruent.
We have the length of leg 1 is equal to the length of leg 2
and the length of the hypotenuse is equal to the length
of the leg times square root of 2.
Now, we put in the variables or values for the quantities
in these two equations.
Leg 1 is x or y, it doesn't matter.
And, the other leg is y. We have two unknowns
and one equation here, so it's not going to get us anywhere.
The hypotenuse is 8 and the leg is x or you can choose to put it
as y if you want to times square root of 2.
In this second equation, we have one unknown, so we can solve
for x by dividing the equation by the square root of 2.
We get x is equal to 8 divided by the square root of 2.
We need to rationalize this number,
because we cannot have radical in the denominator.
And, to rationalize it, we will multiply the fraction
by square root of 2 over square root of 2.
Note that this quantity is just 1.
It doesn't change the value of the original fraction.
So, we have 8 times square root of 2 is 8 square root of 2,
rad 2 times rad 2 is 2.
We can then reduce 8 divided by 2 is 4.
So, x is equal to 4 rad 2 units.
Now, we go back to the first equation.
We already found the value of x, so we just plug it in.
That means, y is equal to 4 rad 2 as well.
And, this completes the problem.
Pause the video and try these problems.
In objective 2, you are going to learn
to determine the side lengths of a second type
of special right triangle, which are the 30,
60, 90 degrees triangles.
Let triangle ABC be a right triangle with measure
of angle A equals 30 degrees and the measure
of angle C equals 60 degrees.
Then, of course, the measure of angle B is 90 degrees,
because it is a right angle.
You have learned that in a triangle,
the side opposite the smallest angle is the shortest side
and the side opposite the largest angle is the
longest side.
Therefore, BC would be the shortest side
because it is opposite the 30 degrees angle.
And, since it is a leg of a right triangle,
so we call it the short leg.
And, the other leg is called the long leg.
Because it is the side opposite the 60 degrees angle,
which is larger than the 30 degrees angle,
and AC is the hypotenuse.
It is the longest side of a right triangle.
Now, suppose that the length of a short leg is some x units,
then the length of the long leg is the length
of the short leg times square root of 3 or it is x rad 3.
And, the length
of the hypotenuse is always 2 times the length
of the short leg or in this case, it is 2x.
This is the special relationship of a 30,
60, 90 degrees triangle.
We can verify this relationship using the Pythagorean theorem.
Short leg squared plus long leg squared should equal
to hypotenuse squared, which is commonly known
as a squared plus b squared is c squared.
The length of the short leg is x,
the length of the long leg is x rad 3, and we want to know
if this is really the hypotenuse squared or 2x squared.
Bring this x squared down.
The square of x is x squared.
The square of rad 3 is 3.
The square of 2x is 4x squared.
X squared times 3 is 3x squared.
And, indeed, x squared plus 3x squared is 4x squared,
which is equal to the right-hand side.
We've verified the special relationship among the sides
of the 30, 60, 90 degrees right triangle.
You should memorize these relationships.
And, we will learn to apply them in the following examples.
In this example, we want to find the values of y and h in a 30,
60, 90 degrees triangle.
The first thing we need to do is to identify the sides.
This is the short leg, because it is opposite the 30
degrees angle.
This is the long leg, because it is opposite the 60
degrees angle.
And, this is the hypotenuse,
because it is opposite the right angle.
Now, we'll use the relationships among the sides in a 30,
60 degrees triangles that we saw earlier,
that is the long leg is equal to the short leg times square root
of 3 and the hypotenuse is 2 times the short leg.
Now, we plug in the variable.
Long leg is y, short leg is 7.
Hypotenuse is h and short leg is 7.
Y is then equal to 7 rad 3 units and h is 14.
This is similar to the previous example.
So, the first thing we need
to do is once again identify the sides in the triangle.
This is the short leg, because it is opposite the 30
degrees angle.
This is the long leg, because it is opposite the 60
degrees angle.
And, this is the hypotenuse.
Now, we write down the equations
of the relationships of the sides.
And, I do recommend that you write these down,
because it will help you in the calculation process.
The long leg is y and the short leg is x. The hypotenuse is 8.
Now, between these two equations,
we see that we can solve for x in the second equation.
So, we divide both sides by 2 and get x is equal to 4.
Now, substitute 4 into x. In this equation,
we get y is equal to 4 rad 3 units.
Once again, we begin by first identifying the sides.
This is the short leg, this is the long leg,
and this is the hypotenuse.
Then, we have this relationship.
Now, we substitute the variables and number in.
The long leg is 5, the short leg is x. The hypotenuse is y
and the short leg is x. Then, we will solve
for x in this equation.
To do so, we divide both sides by the square root of 3.
We get x is equal to 5 over the square root of 3.
We need to rationalize this
because we cannot have radical in the denominator.
To do so, we will multiply top and bottom
by the square root of 3.
Note that square root of 3 over the square root of 3 is 1.
So, the product of 5 over rad 3 times 1 is itself.
Now, we multiply across.
5 times the square root 3 is 5 rad 3.
In the bottom, rad 3 times rad 3 is 3.
This cannot be simplified anymore,
so x is 5 rad 3 over 3 units.
We substitute this value of x into this equation to find y.
We multiply the numerators together,
then we get 2 times 5 is 10.
And, in the bottom, we just get 3.
This cannot be simplified further, therefore,
y is 10 rad 3 over 3 units.
In this example, the hypotenuse is given to us
and we're supposed to find the lengths of the two legs.
X is the short leg, because it is opposite the 30
degrees angle.
Y is the long leg, because it's opposite the 60 degrees angle.
Then, we'll use this relationship once again.
The long leg is y, the short leg is x,
they hypotenuse is 6 square root of 5,
and the short leg is x. Now, we can solve for x in this equation
by dividing both sides by 2.
Now, we can simplify this, 6 divided by 2 we get 3.
So, x is equal to 3 square root of 5.
Now, we substitute this value into x
in this equation to get y.
We can multiply the two radicals together and get 3 square root
of 5 times 3 or it is 3 square root of 15.
And, this is the value of y.
Pause the video and try these problems.