1. **State the problem:** Marlena has three straws with lengths 12 inches, 9 inches, and an unknown shortest straw length $x$. The three straws form an obtuse triangle. We need to find which values among 5, 6, 7, 8, or 9 inches for $x$ produce an obtuse triangle. 2. **Triangle inequalities:** The three sides must satisfy the triangle inequality: $$ x + 9 > 12 \, , \, x + 12 > 9 \, , \, 9 + 12 > x $$ Simplifying: - $x + 9 > 12 \implies x > 3$ - $x + 12 > 9$ always true since $x > 0$ - $9 + 12 > x \implies 21 > x$ So $x$ must be between $3$ and $21$ to form any triangle. 3. **Condition for obtuse triangle:** The triangle is obtuse if the square of the longest side is greater than the sum of the squares of the other two sides. - When $x$ is shortest, the longest side is 12. Check if $12^2 > 9^2 + x^2$: $$144 > 81 + x^2 \implies x^2 < 63 \implies x < \sqrt{63} \approx 7.937$$ - If $x$ is longer than 9, then 12 is still the longest side; same check applies. - If $x$ is between 9 and 12, longest still 12. - If $x$ is larger than 12, longest side is $x$ and check changes. Since $x$ is shortest straw, $x < 9$ 4. **Check each candidate:** - $x=5$: Is $144 > 81 + 25$? $144 > 106$ true, so obtuse. - $x=6$: Is $144 > 81 + 36$? $144 > 117$ true, obtuse. - $x=7$: Is $144 > 81 + 49$? $144 > 130$ true, obtuse. - $x=8$: Is $144 > 81 + 64$? $144 > 145$ false, not obtuse. - $x=9$: If $x=9$, sides are 12, 9, 9 (isosceles). Check if obtuse: $12^2 > 9^2 + 9^2$? $144 > 81+81 = 162$ false, not obtuse. 5. **Conclusion:** Possible shortest straw lengths for obtuse triangle: $5$, $6$, and $7$ inches.