Question
Two sides of a triangle measure 8 cm and 15 cm. Which could be the length of the third side?
6 cm
18 cm
24 cm
28 cm
6 cm
18 cm
24 cm
28 cm
Answers
Answered by
GPT-4o mini
To find the possible lengths of the third side of a triangle when two sides are given, 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.
Given:
- Side a = 8 cm
- Side b = 15 cm
- Side c = the unknown third side
According to the triangle inequality, we have the following conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the values we have:
1. \( 8 + 15 > c \) which simplifies to \( 23 > c \) or \( c < 23 \)
2. \( 8 + c > 15 \) which simplifies to \( c > 7 \)
3. \( 15 + c > 8 \) which simplifies to \( c > -7 \) (this condition is always true since c must be positive)
Now, combining the inequalities from conditions 1 and 2, we have:
\[ 7 < c < 23 \]
Now let's check the options given:
1. **6 cm**: Does not satisfy \( c > 7 \) (not valid)
2. **18 cm**: Satisfies \( 7 < 18 < 23 \) (valid)
3. **24 cm**: Does not satisfy \( c < 23 \) (not valid)
4. **28 cm**: Does not satisfy \( c < 23 \) (not valid)
Thus, the only possible length for the third side is **18 cm**.
Given:
- Side a = 8 cm
- Side b = 15 cm
- Side c = the unknown third side
According to the triangle inequality, we have the following conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the values we have:
1. \( 8 + 15 > c \) which simplifies to \( 23 > c \) or \( c < 23 \)
2. \( 8 + c > 15 \) which simplifies to \( c > 7 \)
3. \( 15 + c > 8 \) which simplifies to \( c > -7 \) (this condition is always true since c must be positive)
Now, combining the inequalities from conditions 1 and 2, we have:
\[ 7 < c < 23 \]
Now let's check the options given:
1. **6 cm**: Does not satisfy \( c > 7 \) (not valid)
2. **18 cm**: Satisfies \( 7 < 18 < 23 \) (valid)
3. **24 cm**: Does not satisfy \( c < 23 \) (not valid)
4. **28 cm**: Does not satisfy \( c < 23 \) (not valid)
Thus, the only possible length for the third side is **18 cm**.
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