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Answer :
To determine the range in which the length of the third side of a triangle must lie, we can use the Triangle Inequality Theorem. This 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.
Let's go through the steps:
1. Label the sides: We have two sides of the triangle: one with length 29 units and another with length 40 units. We'll call the third side [tex]\( x \)[/tex].
2. Apply the Triangle Inequality Theorem:
- For the side [tex]\( x \)[/tex], the theorem gives us three inequalities:
1. [tex]\( 29 + 40 > x \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
3. Simplify the inequalities:
- From [tex]\( 29 + 40 > x \)[/tex], we get [tex]\( x < 69 \)[/tex].
- From [tex]\( 29 + x > 40 \)[/tex], we get [tex]\( x > 11 \)[/tex].
- The third inequality [tex]\( 40 + x > 29 \)[/tex] is always true for [tex]\( x > -11 \)[/tex], so it doesn't provide new information when considering positive lengths.
4. Combine the inequalities:
- The combined result from the simplified inequalities gives us: [tex]\( 11 < x < 69 \)[/tex].
In conclusion, the length of the third side must lie between 11 and 69, but not including 11 and 69 themselves since these inequalities are strict.
Therefore, the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
Let's go through the steps:
1. Label the sides: We have two sides of the triangle: one with length 29 units and another with length 40 units. We'll call the third side [tex]\( x \)[/tex].
2. Apply the Triangle Inequality Theorem:
- For the side [tex]\( x \)[/tex], the theorem gives us three inequalities:
1. [tex]\( 29 + 40 > x \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
3. Simplify the inequalities:
- From [tex]\( 29 + 40 > x \)[/tex], we get [tex]\( x < 69 \)[/tex].
- From [tex]\( 29 + x > 40 \)[/tex], we get [tex]\( x > 11 \)[/tex].
- The third inequality [tex]\( 40 + x > 29 \)[/tex] is always true for [tex]\( x > -11 \)[/tex], so it doesn't provide new information when considering positive lengths.
4. Combine the inequalities:
- The combined result from the simplified inequalities gives us: [tex]\( 11 < x < 69 \)[/tex].
In conclusion, the length of the third side must lie between 11 and 69, but not including 11 and 69 themselves since these inequalities are strict.
Therefore, the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
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