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Answer :
To find 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 denote the sides of the triangle as follows:
- Side 1: 29 units
- Side 2: 40 units
- Side 3: [tex]\( x \)[/tex] units (the length we need to find)
Using the triangle inequality theorem, we set up the following inequalities:
1. [tex]\( 29 + 40 > x \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
Now let's solve each inequality:
1. [tex]\( 29 + 40 > x \)[/tex]
- [tex]\( 69 > x \)[/tex] or [tex]\( x < 69 \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
- Subtract 29 from both sides: [tex]\( x > 11 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
- Subtract 40 from both sides: [tex]\( x > -11 \)[/tex]
Since the requirement from the third inequality [tex]\( x > -11 \)[/tex] is always satisfied when [tex]\( x > 11 \)[/tex], we only need to consider the range from inequalities 1 and 2. Hence, combining these two results, we have:
[tex]\( 11 < x < 69 \)[/tex]
Therefore, the third side must lie strictly between 11 and 69 units. So the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
Let's denote the sides of the triangle as follows:
- Side 1: 29 units
- Side 2: 40 units
- Side 3: [tex]\( x \)[/tex] units (the length we need to find)
Using the triangle inequality theorem, we set up the following inequalities:
1. [tex]\( 29 + 40 > x \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
Now let's solve each inequality:
1. [tex]\( 29 + 40 > x \)[/tex]
- [tex]\( 69 > x \)[/tex] or [tex]\( x < 69 \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
- Subtract 29 from both sides: [tex]\( x > 11 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
- Subtract 40 from both sides: [tex]\( x > -11 \)[/tex]
Since the requirement from the third inequality [tex]\( x > -11 \)[/tex] is always satisfied when [tex]\( x > 11 \)[/tex], we only need to consider the range from inequalities 1 and 2. Hence, combining these two results, we have:
[tex]\( 11 < x < 69 \)[/tex]
Therefore, the third side must lie strictly between 11 and 69 units. So the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
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