To determine the possible lengths of the third side of a triangle given two sides, we 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.
In Ashlyn's case, we have two sides measuring 8 inches and 12 inches. Let's denote the length of the third side as \( x \). The triangle inequality gives us the following three conditions:
- \( 8 + 12 > x \)
- \( 8 + x > 12 \)
- \( 12 + x > 8 \)
Now, let's solve each inequality:
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From \( 8 + 12 > x \): \[ 20 > x \quad \text{or} \quad x < 20 \]
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From \( 8 + x > 12 \): \[ x > 12 - 8 \quad \text{or} \quad x > 4 \]
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From \( 12 + x > 8 \): \[ x > 8 - 12 \quad \text{or} \quad x > -4 \quad \text{(always true since } x \text{ will be positive)} \]
Combining the relevant conditions from inequalities 1 and 2, we find that \( x \) must satisfy: \[ 4 < x < 20 \]
Now we can evaluate the possible lengths for the third piece:
- 1 inch: Not valid (since \( 1 < 4 \))
- 4 inches: Not valid (since \( 4 \) is not greater than \( 4 \))
- 5 inches: Valid (since \( 4 < 5 < 20 \))
- 20 inches: Not valid (since \( 20 \) is not less than \( 20 \))
Therefore, the only possible length for the third piece that would allow Ashlyn to form a triangle is 5 inches.