To determine which side length would allow the creation of a triangle with the two given sides (12.5 inches and 22 inches), we can use the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let’s denote the unknown side length by \( x \), with the other two sides being 12.5 inches and 22 inches. We can apply the triangle inequality conditions as follows:
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\( 12.5 + 22 > x \)
- This simplifies to \( 34.5 > x \) or \( x < 34.5 \)
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\( 12.5 + x > 22 \)
- This simplifies to \( x > 9.5 \)
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\( 22 + x > 12.5 \)
- This simplifies to \( x > -9.5 \) (which is always true for positive \( x \))
Now we combine the results from the first two conditions:
- From \( x < 34.5 \)
- From \( x > 9.5 \)
Thus, \( x \) must satisfy: \[ 9.5 < x < 34.5 \]
Now, let’s evaluate the provided options:
- 4 inches: Not valid (since 4 is less than 9.5)
- 9 inches: Not valid (since 9 is equal to 9.5)
- 11 inches: Valid (since 11 is between 9.5 and 34.5)
- 35 inches: Not valid (since 35 is greater than 34.5)
The only side length from the options given that would create a triangle with the other two sides is 11 inches.
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