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Answer :
To determine the range for the third side of a triangle when you know the lengths of the other two sides, we use the triangle inequality theorem. This theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's apply this to your triangle:
1. You have two sides: 29 units and 40 units.
2. Let's call the third side [tex]\( x \)[/tex].
Now, according to the triangle inequality theorem, we have:
- The sum of the two known sides is greater than the third side:
[tex]\[
29 + 40 > x \implies x < 69
\][/tex]
- The sum of the third side and one known side is greater than the other known side:
[tex]\[
x + 29 > 40 \implies x > 11
\][/tex]
- Similarly:
[tex]\[
x + 40 > 29
\][/tex]
This inequality simplifies to [tex]\( x > -11 \)[/tex] which is always true since a side cannot be negative, so it doesn't affect our problem.
From these inequalities, we conclude that:
- The third side [tex]\( x \)[/tex] must be greater than 11 and less than 69.
Thus, the range of possible lengths for the third side is [tex]\( 11 < x < 69 \)[/tex].
Therefore, the correct answer is option C.
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's apply this to your triangle:
1. You have two sides: 29 units and 40 units.
2. Let's call the third side [tex]\( x \)[/tex].
Now, according to the triangle inequality theorem, we have:
- The sum of the two known sides is greater than the third side:
[tex]\[
29 + 40 > x \implies x < 69
\][/tex]
- The sum of the third side and one known side is greater than the other known side:
[tex]\[
x + 29 > 40 \implies x > 11
\][/tex]
- Similarly:
[tex]\[
x + 40 > 29
\][/tex]
This inequality simplifies to [tex]\( x > -11 \)[/tex] which is always true since a side cannot be negative, so it doesn't affect our problem.
From these inequalities, we conclude that:
- The third side [tex]\( x \)[/tex] must be greater than 11 and less than 69.
Thus, the range of possible lengths for the third side is [tex]\( 11 < x < 69 \)[/tex].
Therefore, the correct answer is option C.
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