Prove a triangle is an acute triangle (Geometry)

You can classify a triangle in 3 different ways.  A triangle can be a right triangle, and acute triangle or an obtuse triangle.  The easiest way to tell if a triangle is either right, acute or obtuse triangle is if you are given the lengths of 3 sides of the triangle, then you can apply the Pythagorean theorem to determine such.  But what if you are only given the length of the longest side of the triangle and just variables for the other 2 sides?  Can you still determine if the triangle is right, acute or obtuse triangle?

In our example, we will use the following dimensions for our triangle.

x, x + 4, 20

You are also told that the side with the length of 20 is the longest side of the triangle.  Can you then prove that this is an acute triangle?  Definitely, we can prove whether it is an acute/obtuse/right triangle.  There are several steps you will need to follow in order to come to the final conclusion.

Step 1:  use the acute triangle (a^2 + b^2 > c^2)  fact to come up with a range for the length of the side.

Since 20 is the longest side of the triangle (given fact from the question), we can setup the acute triangle fact:

a^2 + b^2 > c^2
x^2 + (x + 4)^2 > 20^2
x^2 + (x^2 + 8x + 16) > 400
2x^2 + 8x - 384 > 0
x^2 + 4x - 192 > 0

Using the quadratic formula to solve for the value x, you will obtain the 2 values of x as 12 or -16.  Since the length a triangle cannot be a negative number, we can safely ignore the solution of -16.  This tells us that

x - 12 > 0 or x > 12

Step 2:  Use the triangle inequality theorem to further narrow the range of the lengths of the sides.  Triangle inequality theorem states that the sum of the lengths of the 2 sides of a triangle has to be greater than the length of the third side.  So we can setup the following relationship and solve for the range:

x + (x + 4) > 20
2x + 4 > 20
2x > 16
x > 8

Step 3:  setup a relationship based on the longest side of the triangle versus the second longest side.  Second longest side of the triangle has to be shorter than the longest side.

x + 4 < 20
x < 16

Step 4:  use all the various ranges from step 1, step 2, and step3 and come up with the final range.  The solution MUST satisfy all 3 of the conditions.

Condition #1:  x > 12
Condition #2:  x > 8
Condition #3:  x < 16

The answer that will fit all 3 conditions is

12 < x < 16

This is range for the value of x that will make our initial triangle to be an acute triangle.

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
glenrock@mathnasium.com