Isosceles Right Triangle Assignment

Related Topics:
30-60-90 right triangle
Other special right triangles
More Geometry Lessons



 


Recognizing special right triangles in geometry can provide a shortcut when answering some questions. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem formula, but if you can see that it is a special triangle it can save you some calculations.

In these lessons, we will learn

  • the special right triangle called the 45°-45°-90° triangle
  • how to solve problems involving the 45°-45°-90° right triangle
What is a 45º-45º-90º Triangle?

A 45-45-90 triangle is a special right triangle whose angles are 45º, 45º and 90º. The lengths of the sides of a 45º-45º-90º triangle are in the ratio of 1:1:√2.
The following diagram shows a 45-45-90 triangle and the ratio of its sides. Scroll down the page for more examples and solutions.


Note that a 45°-45°-90° triangle is an isosceles right triangle. It is also sometimes called a 45-45 right triangle.

A right triangle with two sides of equal lengths is a 45°-45°-90° triangle.

You can also recognize a 45°-45°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is 45° then it must be a 45°-45°-90° special right triangle.

A right triangle with a 45° angle must be a 45°-45°-90° special right triangle.

How to solve problems with 45°-45°-90° triangles?

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches.

Solution:

Step 1: This is a right triangle with two equal sides so it must be a 45°-45°-90° triangle.

Step 2: You are given that the both the sides are 3. If the first and second value of the ratio  is 3 then the length of the third side is

Answer: The length of the hypotenuse is  inches.

Example 2:

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is inches and one of the angles is 45°.

Solution:

Step 1: This is a right triangle with a 45° so it must be a 45°-45°-90° triangle.

Step 2: You are given that the hypotenuse is . If the third value of the ratio  is then the lengths of the other two sides must 4.

Answer: The lengths of the two sides are both 4 inches.



Videos

The following videos show more examples of 45-45-90 triangles.

How to find the length of a leg or hypotenuse in a 45-45-90 triangle using the Pythagorean Theorem and then derive the ratio between the length of a leg and the hypotenuse?
This video gives an introduction to the 45-45-90 triangles and shows how to derive the ratio between the lengths of legs and the hypotenuse. How to solve a 45-45-90 triangle given the length of one side by using the ratio?Special Right Triangles in Geometry: 45-45-90 and 30-60-90
This video discusses two special right triangles, how to derive the formulas to find the lengths of the sides of the triangles by knowing the length of one side, and then does a few examples using them. Example problems of finding the sides of a 45-45-90 triangle with answer in simplest radical form.


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.



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Something special in geometry
is the 45, 45, 90 triangle.


Well, a 45, 45, 90 triangle is an isosceles
right triangle where these two legs
are congruent to each other.
The reason why it's 45, 45, 90 is because
if we know that these two angles are
congruent to each other, because the
isosceles triangle theorem, then
we can say that 180 degrees is equal
to 90, plus X plus X. So if
I add these up, I'm going to have 180
is equal to 90, plus 2 X, so I'm
going to subtract 90 from both sides
and I get 90 is equal to 2X, and
then I'm going to divide by 2 to
solve for X. And 90 divided by
2 is 45, which means each of these angles
that are congruent to each other
have to be 45 degrees.


So in an isosceles right triangle you're
going to have a 45 degree, a 45 degree
and a 90 degree.
So that's we mean when
we say 45, 45, 90.


Now something is going on with
these angles and sides.
And if I wrote in that these were both X and
I would say that this is my hypotenuse
C, let's apply the Pythagorean
theorem and see what happens.


Pythagorean theorem says A squared plus
B squared equals C squared and A and
B here are both X. So I'm going to
write that X squared plus X squared
is equal to C squared.
I can combine like terms here and X squared
plus X squared is 2X squared.
So if I want to solve for my hypotenuse
C, I'm going to take the square root
of both sides, and the square root
of X squared is X, and there is no
whole number square root of 2. So
C is equal to X times the square
root of 2. Well, that's a little
difficult to understand.


So let's say we had an isosceles right triangle
with sides of length 1 and I'm
trying to find the hypotenuse.
So maybe this will make sense
with this triangle.


Here we'll have 1 squared plus 1
squared is equal to C squared
Well, 1 plus 1 is 2. So if I take the
square root of both sides, I find
that my hypotenuse is equal to the
square root of 2. So now what
I see it's talking about is if you know
the side of one of your legs, if
you know that length, you're going to.
multiply it by the square root of 2.
So to get from the leg in a 45, 45, 90.
triangle, you're going to multiply by
the square root of 2.


Let's say, however, you don't know what that
leg is. And you know the hypotenuse.
So I'm going to draw another
triangle over here.
45, 45, 90, and let's say you said this
was 3. To go from your hypotenuse
to your leg, you're going to undo multiplying
by the square root of 2.
So you're going to divide by the square
root of 2. So this answer
right here will be 3 divided by the
square root of 2.


And we can't
have a square root in our denominator here.
So now this is becoming quite a chore.
We're going to multiply by square root
of 2. Multiply by the square root
of 2. So we'll have in our numerator
3 times the square root of
2. Square root of 2 times square root of 2
is 2 because you'll have the square root
of 4. So this is actually 3 times.
the square root of 2 divided by 2.


So if we go back to our original
drawing here where we said.
for any right triangle where you have
two legs that are congruent, to go
from your leg to your hypotenuse, all
you need to do is take that number
and multiply it by the square root of
2. So if X is 5, your hypotenuse
is 5 times the square
root of 2. To go from your hypotenuse back to one of
your legs, you're going to divide by
the square root of 2.


So keep that in mind and solving for missing
sides, an isosceles right triangle
is pretty simple.

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