A corner of a fenced yard forms a right angle. Can you place a 12 foot long board across the corner to form a right triangle for which the leg lengths are whole numbers? Explain.

Steps to Solve

1. Understand the Pythagorean theorem The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$). This is expressed as: $$ c^2 = a^2 + b^2 $$

2. Identify the hypotenuse In this case, the 12-foot long board serves as the hypotenuse ($c$) of the right triangle. Thus, we have: $$ c = 12 $$

3. Set up the equation to find leg lengths We need to find whole numbers for legs $a$ and $b$ such that: $$ 12^2 = a^2 + b^2 $$

4. Calculate $12^2$ Calculating the square of 12 gives: $$ 12^2 = 144 $$

5. Rewrite the equation Now the equation becomes: $$ 144 = a^2 + b^2 $$

6. Find integer pairs $(a, b)$ We need to find pairs of whole numbers $(a, b)$ that satisfy the equation. We can use trial and error or organized checking:

• Start with $a = 1$, increasing up to $12$, and solve for $b^2$: $$ b^2 = 144 - a^2 $$

Testing various values:

• For $a = 6$: $$ b^2 = 144 - 6^2 = 144 - 36 = 108 \text{ (not a perfect square)} $$
• For $a = 8$: $$ b^2 = 144 - 8^2 = 144 - 64 = 80 \text{ (not a perfect square)} $$
• For $a = 10$: $$ b^2 = 144 - 10^2 = 144 - 100 = 44 \text{ (not a perfect square)} $$
• For $a = 12$: $$ b = 0 $$

Continue testing until finding values that can form perfect squares.

7. Identify suitable whole number pairs Only $(a, b) = (9, 12)$ and $(12, 9)$ will give perfect squares (e.g., $9^2 + 12^2 = 81 + 144 = 225$, not valid). However, if integer pairs exist, confirm valid combinations.