To all:

I must apologize for a previous post where I was speaking about finding values to make a triangle into an obtuse or acute triangle.

If given 2 sides of a triangle and asked to find a 3rd side such that it is the smallest/largest value to make an acute/obtuse triangle then do the following:

For an obtuse triangle:

For obtuse triangles, finding the smallest/largest value of the 3rd side is easy.  What you do is find the smallest/largest integer value that makes a triangle.  In other words for the smallest value: |a-b|+1, and for the largest value: (a+b)-1.  This will always be an obtuse triangle.

Ex [1]  Find the largest integral value of x, such that 3,7, and x form an obtuse triangle.

• Find 7+3-1 = 9.


• The answer is 9.


• To find the smallest integral value of x, use 7-3+1 = 5.

For an acute triangle:

To find the values of the third side that would make the triangle an acute triangle you can use the following:

sqrt(a^2-b^2)<x<sqrt(a^2+b^2)

So, if you are looking for the smallest integer value of x such that it forms an acute triangle use sqrt(a^2-b^2) (where a is the largest value) and find the highest integer greater than this square root.  If you are looking for the largest value of x, then use sqrt(a^2+b^2) and find the highest integer less than this square root.

Ex [2]  Find the smallest integral value of x such that 5, x, 9 forms an acute triangle.

• 
  Since we are looking for the smallest value we use sqrt(a^2-b^2) = sqrt (9^2-5^2) = sqrt (56).  The highest integer value greater than this is 8.  The answer is 8.


• 
  If we were looking for the largest value, we use sqrt (a^2+b^2) = sqrt (9^2+5^2) = sqrt (106).  The highest integer value less than this is 10.  The answer is 10.

Sorry about the mix up before.  This is the correct way!

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