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Answer :
To determine the range in which the length of the third side of a triangle must lie when the other two sides are given as 29 units and 40 units, we use the Triangle Inequality Theorem.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(x\)[/tex] (the third side), the theorem is expressed as follows:
1. [tex]\( a + b > x \)[/tex]
2. [tex]\( a + x > b \)[/tex]
3. [tex]\( b + x > a \)[/tex]
Let's apply these conditions:
1. [tex]\( 29 + 40 > x \)[/tex]
- This simplifies to [tex]\( 69 > x \)[/tex], so [tex]\( x < 69 \)[/tex].
2. [tex]\( 29 + x > 40 \)[/tex]
- This simplifies to [tex]\( x > 11 \)[/tex].
3. [tex]\( 40 + x > 29 \)[/tex]
- This simplifies to [tex]\( x > -11 \)[/tex] (which doesn't impose any additional restriction since [tex]\( 11 > -11 \)[/tex]).
From conditions 1 and 2, the combined range we find is:
- The length of the third side [tex]\( x \)[/tex] must be greater than 11 and less than 69.
Therefore, the correct range in which the length of the third side must lie is [tex]\( 11 < x < 69 \)[/tex].
Thus, the correct answer is option C.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(x\)[/tex] (the third side), the theorem is expressed as follows:
1. [tex]\( a + b > x \)[/tex]
2. [tex]\( a + x > b \)[/tex]
3. [tex]\( b + x > a \)[/tex]
Let's apply these conditions:
1. [tex]\( 29 + 40 > x \)[/tex]
- This simplifies to [tex]\( 69 > x \)[/tex], so [tex]\( x < 69 \)[/tex].
2. [tex]\( 29 + x > 40 \)[/tex]
- This simplifies to [tex]\( x > 11 \)[/tex].
3. [tex]\( 40 + x > 29 \)[/tex]
- This simplifies to [tex]\( x > -11 \)[/tex] (which doesn't impose any additional restriction since [tex]\( 11 > -11 \)[/tex]).
From conditions 1 and 2, the combined range we find is:
- The length of the third side [tex]\( x \)[/tex] must be greater than 11 and less than 69.
Therefore, the correct range in which the length of the third side must lie is [tex]\( 11 < x < 69 \)[/tex].
Thus, the correct answer is option C.
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