Question 1203966: What is the lowest whole-number side length (in inches) that would ensure the triangle is an obtuse triangle?
A triangle with side lengths of 12 inches, 16 inches, and blank inches.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52884)   (Show Source):
You can put this solution on YOUR website! .
Third side of the triangle must be longer than 16-12 = 4 inches.
In whole number of inches, the minimum length of the third side of the triangle is 5 inches.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Review this triangle inequality lesson
https://www.algebra.com/algebra/homework/Triangles/triangle-inequality-theorem.lesson
Focus on Example 3.
I'll follow that outline for this current triangle.
We have a triangle with sides: a,b,c
where,
a = 12
b = 16
c = unknown
Then,
b-a < c < b+a
16-12 < c < 16+12
4 < c < 28
If c is an integer, then the possible values would be {5,6,7,...,25,26,27}
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If c is the longest side and if  is the case, then we have an obtuse triangle.
Search out "converse of pythagorean theorem" for more info.
a = 12
b = 16
If c = 20, then  to give us a right triangle with sides 12,16,20
If c > 20, then we'll meet the condition that
Example: c = 21 leads to  which exceeds
Therefore, a triangle with sides 12,16,21 is obtuse.
We have determined that c = 21 is the smallest integer value possible for c when c is the longest side.
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Now we must consider the case when c isn't the longest side.
The longest side is now 16 units.
a = 12
b = 16 ... longest side
c = some value on the interval 4 < c < 16
With b as the longest side and  i.e.  , then the triangle is obtuse. I'm using a similar template as the previous section.
b^2 - a^2 = 16^2 - 12^2 = 112
c^2 < 112
c < sqrt(112)
c < 10.583005
c < 11 when c is an integer
If c isn't the longest side, then c should take on these integer values {5,6,7,8,9,10} to produce an obtuse triangle.
Therefore, c = 5 gives the lowest whole-number side length for the missing side, such that this triangle is obtuse.
This interactive GeoGebra applet let's you see various possibilities.
https://www.geogebra.org/m/er4ypqxc
If the canvas is blank, you may need to refresh the page.
Or you may need to click the curved arrow at the center of the screen (after hovering your mouse over top).
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Answer: 5
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