To find the range of possible values for the third side of a triangle when you know the lengths of the other two sides, you can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given two sides of lengths \( a = 15 , \text{ft} \) and \( b = 22 , \text{ft} \), we denote the third side as \( c \). The triangle inequality gives us three inequalities to satisfy:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Plugging in the values we have:
- \( 15 + 22 > c \) → \( 37 > c \) → \( c < 37 \)
- \( 15 + c > 22 \) → \( c > 7 \)
- \( 22 + c > 15 \) → \( c > -7 \) (this inequality is always true for positive side lengths)
Now we combine \( c > 7 \) and \( c < 37 \):
Thus, the range for the third side \( c \) is:
\[ 7 < c < 37 \]
Among the options provided, the correct answer is \( 7 < x < 37 \).
1 answer