Question 1200226
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Answers:
<font color=red size=4>4 cm
10 cm
18 cm</font>


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Explanation:


We have a triangle with sides a,b, and c.


a = 9 and b = 12 are the two known sides.
c is the unknown side.


Due to the <a href = "https://www.mathsisfun.com/geometry/triangle-inequality-theorem.html">triangle inequality theorem</a>, we have these three conditions that must all be true if we want a triangle to be possible
a < b+c
b < a+c
c < a+b
Rephrased verbally: "The sum of any two sides must exceed the third side".


Let's isolate c in the 1st two inequalities
a < b+c becomes a-b < c
b < a+c becomes b-a < c


If b ≥ a, then we'll focus on b-a < c so that the left hand side is nonnegative.


At the same time, the third inequality condition of the triangle inequality theorem states that c < a+b or c < b+a


We have these inequalities
b-a < c and c < b+a
they can be conveniently glued together to get
<font color=blue>b-a < c < b+a</font>
This gives a range of possible c values where b ≥ a.


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Summary of the previous section:


Given a triangle with sides {a,b,c} where a & b are known lengths, and b ≥ a, the third side c has the restrictions of: <font color=blue>b-a < c < b+a</font>


From here we plug in a = 9 and b = 12
b-a < c < b+a
12-9 < c < 12+9
3 < c < 21


The third side is between 3 cm and 21 cm, excluding each endpoint. 
We cannot have c = 3. We also cannot have c = 21. 
These endpoints will cause a straight line to form rather than a triangle.


The following answer choices are in the interval mentioned: 
<font color=red size=4>4 cm
10 cm
18 cm</font>
which are the final answers.
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