Two sides of a triangle are 11 and 17. How many possible lengths are there for the third side, if it is a positive integer?
I tried listing them all but I get confused...Please help thank you!
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The sum of any two sides must be greater than the third side for a triangle to exist
let the third side be x
x+11>17 AND x+17>11 AND 11+17> x
x > 6 AND x>-6 AND x < 28
so 6 < x < 28
So how many positive integers can you count between 6 and 28 exclusive ?
Why did the triangle go to therapy? Because it couldn't find its third side!
But don't worry, I'm here to help! To find the possible lengths for the third side of a triangle, we need to consider the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, let's take the two given sides: 11 and 17. To find the possible lengths for the third side, we need to check which lengths satisfy the triangle inequality theorem.
So, we can check the following inequalities:
11 + 17 > third side
17 + third side > 11
11 + third side > 17
From here, we can simplify the inequalities:
28 > third side
17 + third side > 11
11 + third side > 17
Now, let's solve these inequalities step by step:
For the first one, third side can be any positive integer less than 28.
For the second and third inequalities, we can simplify them further:
third side > -6
third side > 6
Since we are looking for positive integers as the third side, we can disregard the first inequality (third side > -6). Therefore, the possible lengths for the third side are any positive integers greater than 6.
So, there are infinitely many possible lengths for the third side of the triangle, as long as it is a positive integer greater than 6.
Hope that clears things up for you! Keep enjoying triangles, they're a real "angle" to work with!
To find the possible lengths for the third side of a triangle, we need to apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, the two given sides are 11 and 17. Let's denote the length of the third side as x. Using the theorem, we can set up the following inequality:
11 + 17 > x
Since x must be a positive integer, we can simplify the inequality to:
28 > x
Therefore, the possible lengths for the third side are all the integers greater than 0 but less than 28. To find the exact number of possible lengths, we subtract 1 from 28 to exclude the upper limit of 28:
Number of possible lengths = 27
So, there are 27 possible lengths for the third side of the triangle.
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Let the third side be $n$. Then by the triangle inequality,
\begin{align*} n + 11 &> 17, \\ n + 17 &> 11, \\ 11 + 17 &> n, \end{align*}
which gives us the inequalities $n > 6$, $n > -6$, and $n < 28$. Therefore, the possible values of $n$ are 7, 8, $\dots$, 27, for a total of $27 - 7 + 1 = \boxed{21}$ possible values.