<PRE><B>Question 1203040</B>
&lt;br&gt;
In a right triangle with the standard notation with c being the length of the longest side (hypotenuse), the relationship {{{c^2=a^2+b^2}}} holds.&lt;br&gt;
For a triangle to be obtuse, again using c to denote the length of the longest side, the relationship {{{c^2&gt;a^2+b^2}}} must hold.&lt;br&gt;
In this problem, there are two cases to consider -- the longest side can be either k or 15.&lt;br&gt;
(1) k is the longest side&lt;br&gt;
The triangle inequality tells us that k&lt;11+15=26.&lt;br&gt;
For the triangle to be obtuse, we need to have&lt;br&gt;
{{{k^2&gt;11^2+15^2}}}
{{{k^2&gt;346}}}
{{{k&gt;18.6}}} approximately&lt;br&gt;
The integers greater than 18.6 and less than 26 are 19 through 25 inclusive -- that&#39;s 7 possible integer values for k.&lt;br&gt;
(2) 15 is the longest side&lt;br&gt;
Now the triangle inequality tells us k&gt;15-11=4.&lt;br&gt;
For the triangle to be obtuse, we need to have&lt;br&gt;
{{{15^2&gt;k^2+11^2}}}
{{{225-121&gt;k^2}}}
{{{k^2&lt;104}}}
{{{k&lt;10.2}}} approximately&lt;br&gt;
The integers greater than 4 and less than 10.2 are 5 through 10 inclusive -- that&#39;s 6 possible integer values for k.&lt;br&gt;
ANSWER: 7+6 = 13&lt;br&gt;
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