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Sure! To determine which set of two sides is possible for the lengths of the other two sides of a triangle where one side is given as 13 cm, we need to ensure that the triangle inequality theorem is satisfied. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
We’ll test each set of side lengths one by one against the triangle inequality theorem. Let's denote the known side of the triangle as [tex]\( a = 13 \)[/tex] cm.
### Check the sides [tex]\( \boldsymbol{5 \ \text{cm} \ \text{and} \ 8 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 5 > 8 \)[/tex]
- [tex]\( 18 > 8 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 5 + 8 > 13 \)[/tex]
- [tex]\( 13 > 13 \)[/tex] (False)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 8 > 5 \)[/tex]
- [tex]\( 21 > 5 \)[/tex] (True)
Since one of the inequalities is not satisfied, [tex]\( 5 \ \text{cm} \ \text{and} \ 8 \ \text{cm} \)[/tex] is not a valid set.
### Check the sides [tex]\( \boldsymbol{6 \ \text{cm} \ \text{and} \ 7 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 6 > 7 \)[/tex]
- [tex]\( 19 > 7 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 6 + 7 > 13 \)[/tex]
- [tex]\( 13 > 13 \)[/tex] (False)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 7 > 6 \)[/tex]
- [tex]\( 20 > 6 \)[/tex] (True)
Since one of the inequalities is not satisfied, [tex]\( 6 \ \text{cm} \ \text{and} \ 7 \ \text{cm} \)[/tex] is not a valid set.
### Check the sides [tex]\( \boldsymbol{7 \ \text{cm} \ \text{and} \ 2 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 7 > 2 \)[/tex]
- [tex]\( 20 > 2 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 7 + 2 > 13 \)[/tex]
- [tex]\( 9 > 13 \)[/tex] (False)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 2 > 7 \)[/tex]
- [tex]\( 15 > 7 \)[/tex] (True)
Since one of the inequalities is not satisfied, [tex]\( 7 \ \text{cm} \ \text{and} \ 2 \ \text{cm} \)[/tex] is not a valid set.
### Check the sides [tex]\( \boldsymbol{8 \ \text{cm} \ \text{and} \ 9 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 8 > 9 \)[/tex]
- [tex]\( 21 > 9 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 8 + 9 > 13 \)[/tex]
- [tex]\( 17 > 13 \)[/tex] (True)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 9 > 8 \)[/tex]
- [tex]\( 22 > 8 \)[/tex] (True)
Since all three inequalities are satisfied, [tex]\( 8 \ \text{cm} \ \text{and} \ 9 \ \text{cm} \)[/tex] is a valid set.
Therefore, the set of two sides that is possible for the lengths of the other two sides of this triangle is [tex]\( 8 \ \text{cm} \ \text{and} \ 9 \ \text{cm} \)[/tex].
We’ll test each set of side lengths one by one against the triangle inequality theorem. Let's denote the known side of the triangle as [tex]\( a = 13 \)[/tex] cm.
### Check the sides [tex]\( \boldsymbol{5 \ \text{cm} \ \text{and} \ 8 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 5 > 8 \)[/tex]
- [tex]\( 18 > 8 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 5 + 8 > 13 \)[/tex]
- [tex]\( 13 > 13 \)[/tex] (False)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 8 > 5 \)[/tex]
- [tex]\( 21 > 5 \)[/tex] (True)
Since one of the inequalities is not satisfied, [tex]\( 5 \ \text{cm} \ \text{and} \ 8 \ \text{cm} \)[/tex] is not a valid set.
### Check the sides [tex]\( \boldsymbol{6 \ \text{cm} \ \text{and} \ 7 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 6 > 7 \)[/tex]
- [tex]\( 19 > 7 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 6 + 7 > 13 \)[/tex]
- [tex]\( 13 > 13 \)[/tex] (False)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 7 > 6 \)[/tex]
- [tex]\( 20 > 6 \)[/tex] (True)
Since one of the inequalities is not satisfied, [tex]\( 6 \ \text{cm} \ \text{and} \ 7 \ \text{cm} \)[/tex] is not a valid set.
### Check the sides [tex]\( \boldsymbol{7 \ \text{cm} \ \text{and} \ 2 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 7 > 2 \)[/tex]
- [tex]\( 20 > 2 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 7 + 2 > 13 \)[/tex]
- [tex]\( 9 > 13 \)[/tex] (False)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 2 > 7 \)[/tex]
- [tex]\( 15 > 7 \)[/tex] (True)
Since one of the inequalities is not satisfied, [tex]\( 7 \ \text{cm} \ \text{and} \ 2 \ \text{cm} \)[/tex] is not a valid set.
### Check the sides [tex]\( \boldsymbol{8 \ \text{cm} \ \text{and} \ 9 \ \text{cm}} \)[/tex]:
1. [tex]\( a + b > c \)[/tex] ⇔ [tex]\( 13 + 8 > 9 \)[/tex]
- [tex]\( 21 > 9 \)[/tex] (True)
2. [tex]\( b + c > a \)[/tex] ⇔ [tex]\( 8 + 9 > 13 \)[/tex]
- [tex]\( 17 > 13 \)[/tex] (True)
3. [tex]\( a + c > b \)[/tex] ⇔ [tex]\( 13 + 9 > 8 \)[/tex]
- [tex]\( 22 > 8 \)[/tex] (True)
Since all three inequalities are satisfied, [tex]\( 8 \ \text{cm} \ \text{and} \ 9 \ \text{cm} \)[/tex] is a valid set.
Therefore, the set of two sides that is possible for the lengths of the other two sides of this triangle is [tex]\( 8 \ \text{cm} \ \text{and} \ 9 \ \text{cm} \)[/tex].
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