Question 945995: given two sides of a triangle 5 and 10, find the following,
range of possible side lengths for triangle to be acute
to be right
to be obtus
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  = length of the third side.
To be a triangle, it must be
 ---> 
Within that range,
very short and very long third sides yield  triangles;
there are values that yield right triangle, amd
in between those values we get acute triangles:
 or  --> obtuse triangle
 or  --> right triangle
 --> acute triangle
For  , we have an isosceles triangle,
with the vertex angle being the smallest angle, with measure  ,
because it is opposite the shortest side.
The other two angles are the base angles,
each measuring  ,
so they are acute too, and we have an acute triangle.
For The values that yield right triangles, we use the Pythagorean theorem.
For  :
We would get a right triangle with  and  legs if and only if
 )
 -->
If is the longest side length, ,
the largest angle would be opposite that longest side.
That angle would be greater than a right angle (an obtuse angle),
if and only if  .
So, we would get an obtuse triangle if and only if .
Also with being the longest side length, ,
the largest angle, opposite that side, would be less than a right angle (acute) if and only if 
So, we would get an acute triangle if  ,
For  :
We would get a right triangle with leg and hypotenuse if and only if
 --->
 ---> .
When the longest side is the one with length 10,
the angle opposite that side is the greatest angle.
If  ,
we would have  :
the greatest angle would be obtuse and the triangle would be obtuse.
If  ,
we would have  :
the greatest angle would be acute and the triangle would be acute.
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