Given two sides of a triangle with lengths 7 ft and 13 ft, find the range of possible lengths for the third side.
Understand the Problem
The question asks to find the possible range of lengths for the third side of a triangle, given the lengths of the other two sides. This involves using the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Answer
$7 < x < 27$
Answer for screen readers
$7 < x < 27$
Steps to Solve
- Apply the Triangle Inequality Theorem
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 third side. Let the length of the third side be $x$. We can set up three inequalities:
$10 + 17 > x$ $10 + x > 17$ $17 + x > 10$
- Solve the first inequality
$10 + 17 > x$ simplifies to $27 > x$, or $x < 27$.
- Solve the second inequality
$10 + x > 17$ simplifies to $x > 17 - 10$, which means $x > 7$.
- Solve the third inequality
$17 + x > 10$ simplifies to $x > 10 - 17$, which means $x > -7$. Since side lengths must be positive, this inequality doesn't give us any additional useful information because we already found that $x > 7$.
- Combine the inequalities
We have $x < 27$ and $x > 7$. Therefore, the length of the third side, $x$, must be between 7 and 27.
- Express the answer as a compound inequality
The possible values for $x$ are $7 < x < 27$.
$7 < x < 27$
More Information
The length of the third side must be strictly greater than 7 and strictly less than 27. It cannot be equal to 7 or 27.
Tips
A common mistake is to only consider one or two of the inequalities derived from the Triangle Inequality Theorem, which leads to an incomplete or incorrect range for the third side. Also, forgetting that side lengths must be positive can lead to errors.
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