SOLUTION: 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

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.
RELATED QUESTIONS
Which of the following cannot be possible to construct a triangle with the given side... (answered by stanbon)

the lengths of two sides of a triangle are 3 in. and 8 in. Find the range of possible... (answered by richwmiller)

The lengths of two sides of a triangle are 13.5 cm and 4.5 cm. Find the range of... (answered by ewatrrr)

two sides of a triangle measure 11 meters and 24 meters. Write an inequality to show the... (answered by Alan3354)

two sides of a triangle have lengths 6 and 11. the third side has to... (answered by oberobic)

1. The lengths of two sides of a triangle are 4 and 9 the length of the third side must... (answered by josgarithmetic)

If a triangle has a side 5 inches long and a second side 10 inches long what is the... (answered by jim_thompson5910)

An acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation (answered by solver91311)

The lengths of two sides of a triangle are 11 and 23. If the third side is x, find the... (answered by reviewermath)