Question 1203040: Please help me with this
The sides of a triangle have lengths 11, 15, and k where k is an integer. For how many values of k, is
the triangle obtuse?
Found 3 solutions by math_helper, ikleyn, greenestamps:
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website!
  
This amounts to 13 values of k.
Go to this site:
https://www.triangle-calculator.com/?what=sss&a=15&b=11&c=19&submit=Solve
Plug in 11,15 as two sides and then 'experiment' with other values for the 3rd side. You will find integer values 5-10 and 19-25 all form obtuse triangles. The site will draw the triangles so you can see where the obtuse angle is.
Best wishes.
.. .. ..
Another approach is to pick a k such that you have a valid triangle. This means the triangle inequalities must each hold for the value of k. For this problem: 11+15>k, k+11>15, and k+15>11. Then, for each triple (11,15,k) pick the longest side and compute the square of each side-length (121,225,k^2). Call the longest side 'c'. If c^2 is greater than the sum of the other two squared values, you have an obtuse triangle.
Example:
Say k=19 ... the triangle inequalities hold, check.
The values of the squares of each side-lengths are: (121,225,361)
Since 361 > 225+121, the triangle is obtuse.
Another example:
Say k=12... the triangle inequalities hold, check.
Squared side-lengths are (121,225,144)
However, since 225 < 121+144, we do NOT have an obtuse triangle.
Answer by ikleyn(52814)  (Show Source):
You can put this solution on YOUR website! .
Please help me with this
The sides of a triangle have lengths 11, 15, and k where k is an integer.
For how many values of k, is the obtuse triangle?
~~~~~~~~~~~~~~~
The solution by tutor @math_helper is fine.
In this post, I bring another solution.
First, from the triangle inequalities, we have these two inequalities for k
k < 11+15 = 26 and k > 15-11 = 4.
So, the possible integer values of k are in this interval 5 <= k <= 25. (1)
Now we will analyze what restrictions come from the condition that the triangle is obtuse.
If the obtuse angle is between the sides 11 and 15, then this inequality must held
k >=  = 18.6 (approximately),
which for integer values of k gives k >= 19.
Thus, combining it with (1), in this case, it must be 19 <= k <= 25. (2)
If the obtuse angle is attached to one of the sides 11 or 15, then this inequality must held
k <=  = 10.2 (approximately),
which for integer values of k gives k <= 10.
Thus, combining it with (1), in this case, it must be 5 <= k <= 10. (3)
From (2) and (3), the final answer is { 5 <= k <= 10 or 19 <= k <= 25 },
giving 13 possible integer values of "k".
Solved.
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Tutor @greenestamps successfully retold my solution to you in his own words for your better understanding.
Answer by greenestamps(13203)  (Show Source):
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