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Answer :
To find the possible lengths for the third side of the triangle, 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 two known sides of the triangle as 7 and 15.
1. First Condition:
- The sum of the lengths of the two known sides must be greater than the third side.
- So, [tex]\(7 + 15 > x\)[/tex].
- Simplifying this, we get:
[tex]\[ x < 22 \][/tex]
2. Second Condition:
- The sum of the third side and one of the known sides must be greater than the other known side.
- So, for the side of length 15:
[tex]\[ 7 + x > 15 \][/tex]
- Simplifying this, we get:
[tex]\[ x > 8 \][/tex]
3. Third Condition:
- The sum of the third side and the other known side must be greater than the remaining known side.
- So, for the side of length 7:
[tex]\[ 15 + x > 7 \][/tex]
- This condition is always true and doesn't add any new constraint since adding a positive number (the third side) to 15 will always be greater than 7.
Combining the two main conditions [tex]\( x > 8 \)[/tex] and [tex]\( x < 22 \)[/tex], we have:
[tex]\[ 8 < x < 22 \][/tex]
So, the correct inequality representing the possible lengths for the third side is:
d. [tex]\(8 < x < 22\)[/tex]
Let's denote the two known sides of the triangle as 7 and 15.
1. First Condition:
- The sum of the lengths of the two known sides must be greater than the third side.
- So, [tex]\(7 + 15 > x\)[/tex].
- Simplifying this, we get:
[tex]\[ x < 22 \][/tex]
2. Second Condition:
- The sum of the third side and one of the known sides must be greater than the other known side.
- So, for the side of length 15:
[tex]\[ 7 + x > 15 \][/tex]
- Simplifying this, we get:
[tex]\[ x > 8 \][/tex]
3. Third Condition:
- The sum of the third side and the other known side must be greater than the remaining known side.
- So, for the side of length 7:
[tex]\[ 15 + x > 7 \][/tex]
- This condition is always true and doesn't add any new constraint since adding a positive number (the third side) to 15 will always be greater than 7.
Combining the two main conditions [tex]\( x > 8 \)[/tex] and [tex]\( x < 22 \)[/tex], we have:
[tex]\[ 8 < x < 22 \][/tex]
So, the correct inequality representing the possible lengths for the third side is:
d. [tex]\(8 < x < 22\)[/tex]
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