Working with special right triangles (Geometry)

Invariably, you will be working with some special right triangles.  Specifically you will be asked to find the lengths of the a 30-60-90 or 45-45-90 triangles given one of the sides.


What students should be aware of is that they have to use their cumulative knowledge to work on this subject area.  Previously students are made aware of the side-angle relationship between a triangle.  Specifically students learned that "If one angle of a triangle has a greater degree measure than another angle, then the side opposite the greater angle will be longer than the side opposite the smaller angle."  Given that fact, the students should know that in a 30-60-90 triangle, the shortest side of the triangle is opposite from 30 degrees, the second longest side is opposite from the 60 degrees angle, and the longest side (also known as the hypotenuse) is directly opposite from the 90 degrees angle.

In order for the student to truly understand this concept, they should think of the shortest side of the special right triangles as the variable x.  Then we setup a relationship for the other sides of the triangles based on shortest side of the triangle.

30-60-90 degree triangle:
Side across from 30 degrees has length of x
Side across from 60 degrees has length of sqrt(3)*x
Side across from 90 degrees has length of 2x

45-45-90 degree triangle:
Side across from 45 degrees has length of x
Side across from 90 degrees has length of sqroot(2) * x

Let's use some concrete example to demonstrate this:

Example 1 (easy):  30-60-90 triangle ABC.  Angle A is 30 degrees, angle B is 60 degrees, and angle C is 90 degrees.  Segment BC (opposite from angle A) has a length of 5.  What is the length of AB (opposite from angle C) and length of AC (opposite from angle B)?

Since angle A is 30 degrees and the side opposite to it has a length of 5.  This means that 5 is shortest length of this triangle.  Remember that the 2nd longest side is across from the 60 degrees, the length of this side is the shortest side multiply by sqrt(3).  So the side across from 60 degrees has a length of 5sqrt(3).  The longest side (hypotenuse) has a length of 2 times the shortest side.  This means that the hypotenuse has a length of 2x5 = 10.

Example 2 (difficult):  30-60-90 triangle ABC.  Angle A is 30 degrees, angle B is 60 degrees, and angle C is 90 degrees.  Segment AB (opposite from angle C) has a length of 7.  What is the length of BC (opposite from angle A) and length of AC (opposite from angle B)?

This example is different from the 1st one because we are not given the shortest side of the triangle.  But as long as we know the relationship between all the sides, we should be able to setup an equation to solve for the sides of the triangle.  Since we are given the longest side (hypotenuse) of the triangle as 7.  We can set the equation as (longest side (AKA hypotenuse) = 2 times the shortest side)

2x = 7
x = 3.5 (segment BC)

Since we know the shortest side is 3.5, we just apply the formula to the 2nd longest side.  So the second longest side is 3.5sqrt(3) (segment AC).

Example 3 (difficult):  45-45-90 triangle ABC.  Angle A is 45 degrees, angle B is 45 degrees, and angle C is 90 degrees.  Segment AB (opposite from angle C) has a length of 4sqrt(2).  What is the length of BC (opposite from angle A) and length of AC (opposite from angle B)?

We should setup this type of question just like before.  What we have been given is the length of the longest side (hypotenuse), so let's take advantage of the side-angle relationship for the 45-45-90 triangle:

sqrt(2) * x = 4sqrt(2)
x = 4

So the lengths of the 2 sides of the triangles opposite to the 45 degrees triangle are 4 each.

If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center.

Michael Huang
Mathnasium of Glen Rock/Ridgewood
236 Rock Road
Glen Rock, NJ 07452
Tel:  201-444-8020
glenrock@mathnasium.com