1. **State the problem:** Emily breaks a stick into 3 pieces that can form a triangle. She loses one piece and measures the other two as 6 cm and 12 cm. We need to find which total length of the original stick is possible from the given options. 2. **Recall the triangle inequality rule:** For any triangle with sides $a$, $b$, and $c$, the sum of any two sides must be greater than the third side: $$a + b > c, \quad b + c > a, \quad c + a > b$$ 3. **Let the three pieces be $6$, $12$, and $x$ cm.** The original stick length is: $$6 + 12 + x = 18 + x$$ 4. **Apply the triangle inequalities:** - $6 + 12 > x \Rightarrow 18 > x$ - $6 + x > 12 \Rightarrow x > 6$ - $12 + x > 6 \Rightarrow x > -6$ (always true since length is positive) So, $6 < x < 18$. 5. **Check each total length option:** - Total length $= 18 + x$ - Since $x$ must be between 6 and 18, total length must be between: $$18 + 6 = 24 \quad \text{and} \quad 18 + 18 = 36$$ 6. **Evaluate options:** - A) 42 cm (too large, > 36) - B) 23 cm (too small, < 24) - C) 20 cm (too small, < 24) - D) 30 cm (between 24 and 36, possible) - E) 37 cm (too large, > 36) **Final answer:** The original stick length can be 30 cm.