TWO SPECIAL TRIANGLES

Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!

Dr. Carol J.V. Fisher's Homepage

For this exercise, you need ♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥

Please go to the updated version of this web exercise:
Two Special Triangles

TWO SPECIAL TRIANGLES

Jump right to the exercises!

You should know the relationship between the lengths of the sides for two special triangles: 30°-60°-90° and 45°-45°-90° .

First, consider a 30°-60°-90° triangle.
The side opposite the 30° angle is shortest.
The hypotenuse is twice as long as the shortest side.
The remaining side (the one opposite the 60° angle) is 3 times as long as the shortest side.

It is very easy to see that these are the correct relationships between the lengths of the sides, as follows:
Start with an equiangular (hence equilateral) triangle, where each side has length 2.
Notice that all the angles must be 180°3=60° .
Drop a perpendicular, as shown below.

The purple and green triangles are 30°-60°-90° triangles, and the length of the shortest side is 1 (why?).
A quick application of the Pythagorean Theorem shows that the remaining side has length 3 .
Finally, scale all sides by s to get the following result:

Lengths of Sides in a 30°-60°-90° Triangle

Let s denote the shortest side in a 30°-60°-90° triangle.
Then, the hypotenuse has length 2s ,
and the side opposite the 60° angle has length 3s .

Conversely, if a triangle has sides of lengths s , 3s and 2s ,
then it is a 30°-60°-90° triangle.

Since 3 is about 1.7 , it follows that the side opposite the 60° angle is a little more than one and a half times the shortest side.

The relationships between the sides in a 45°-45°-90° triangle is even easier.
Create a triangle with two sides of length 1 and a 90° angle, as shown below:

a 45-45-90 triangle

Since angles opposite equal sides must be equal, and since the two remaining angles must add to 90° , they must each be 45° .
A quick application of the Pythagorean Theorem shows that the remaining side must equal 2 .

Finally, scale all sides by s to get the following result:

Lengths of Sides in a 45°-45°-90° Triangle

Let s denote the length of the two shorter sides in a 45°-45°-90° triangle.
Then, the hypotenuse has length 2s .

Conversely, if a triangle has sides of lengths s , s , and 2s then it is a 45°-45°-90° triangle.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.