Posted by **Knights** on Monday, December 31, 2012 at 11:58am.

Two sides of an obtuse triangle are 16 and 21. How many possible lengths are there for the third side, if it is a positive integer?

i know that in an obtuse triangle

a^2+b^2<c^2

a+b>c

but i tried plugging in and it wont work so couldyou guys help me?

btw merry christmas and happy new year

- geometry inequality triangles help if you can -
**Reiny**, Monday, December 31, 2012 at 1:35pm
First let's establish the domain for the third side

let the third side be c

c+16>21 ----> c > 5

c+21 > 16 ----> c > -5 , obviously

21 + 16 > c ---> c < 37

so to be a triangle in the third side must be

5 < c < 37

Now to be obtuse

c^2> 21^2+16^2

c^2 > 697

c > 26

could one of the other angles be obtuse ?

21^2 >c^2 + 16^2

c^2 < 185

c < 14

16^ > c^2 + 21^2

c^2 < a negative, which is not possible

So for the angle opposite the third side to be obtuse,

26 < c < 37

For the angle opposite the side 21 to be obtuse,

5 < c < 14

- geometry inequality triangles help if you can -
**Knights**, Monday, December 31, 2012 at 1:39pm
thanks a lot the answer is 18 right?

merry christmas and happy new year to y

- geometry inequality triangles help if you can -
**Reiny**, Monday, December 31, 2012 at 2:08pm
No, 18 does not work.

Notice my two possible solutions, .....

So for the angle opposite the third side to be obtuse,

26 < c < 37

So the third side could be 27, 28, 29, 30 31 32 33 34 35 or 36

For the angle opposite the side 21 to be obtuse,

5 < c < 14

so the third side could be 6 7 8 9 10 11 12 or 13

make a sketch using one of the answers and illustrate that the angle must be obtuse by using the cosine law.

Also test it with your 18 to show it does not work

- geometry inequality triangles help if you can -
**Knights**, Monday, December 31, 2012 at 9:30pm
I meant the total number of possibilities. Thanks anyway!

## Answer this Question

## Related Questions

- math problem triangle inequality help - Two sides of a triangle are 11 and 17. ...
- Triangle inequality? - Two altitudes of a triangle have lengths 12 and 14. What ...
- Math - The sum of the lengths of any two sides of a triangle must be greater ...
- geometry - Two sides of a triangle have lengths 8 and 17. Which inequalities ...
- Maths - Two sides of a triangle are 10 cm and 15 cm. What is the range of ...
- A Geometry ? - Given that two sides of a triangle are 20 and 66. Find the range ...
- IRSC - the sum of the lengths of any two sides of a triangle must be greater ...
- Math - If all of the following triangles have the same perimeter, which has the ...
- Geometry - In a triangle with integer side lengths, one side is three times as ...
- math - the lengths of the three sides of a right triangle R have odd integer ...