Problem 6 · AMC 8 Stretch
Core
Counting & Probability
Geometry & Measurement
account-for-all-possibilitiesorganizing-datalogical-reasoning
A 9-inch piece of wire is bent at two of the inch marks so its two ends meet, forming a triangle. The two bends must land exactly on inch marks. How many different choices of bending points are possible?
Show answer
Answer: 10 choices
Show hints
Hint 1 of 4
The three sides are whole numbers that add to 9. Which whole-number triples can actually form a triangle? (Triangle rule: the two shorter sides together must be LONGER than the longest side.)
Still stuck? Show hint 2 →
Hint 2 of 4
List the valid shapes: 3-3-3, 1-4-4, and 2-3-4. But careful — the question asks for bending-POINT choices, not just shapes.
Still stuck? Show hint 3 →
Hint 3 of 4
Organize by the first (smaller) bend. If you bend first at 1, then 2, then 3, then 4, how many valid second bends does each allow?
Show solution
Approach: Organize the bending-point pairs by the smaller bend
- Whole-number sides adding to 9 that obey the triangle rule are only three shapes: 3-3-3 (equilateral), 1-4-4 (isosceles), and 2-3-4 (scalene). Many people stop and answer '3' — but the question asks for bending-point choices.
- Lay the wire out as marks 1 through 8; picking two bends fixes where each side falls. List by the smaller bend: bend at 1 and 5 (1 way); bend at 2 and {5 or 6} (2 ways); bend at 3 and {5,6,7} (3 ways); bend at 4 and {5,6,7,8} (4 ways).
- Total: \(1 + 2 + 3 + 4 = 10\) choices.
- So there are 10 choices of bending points. (Notice 10 is the 4th triangular number.)
Mark:
· log in to save