Convert 0.45 to a fraction in simplest form. Which set of lengths forms a right triangle? A. 5, 6, 7 B. 6, 8, 10 C. 3, 4, 8 D. 4, 5, 6

Problem: Convert $0.45$ to a fraction in simplest form and identify which set of lengths $\{5, 6, 7\}$, $\{6, 8, 10\}$, $\{3, 4, 8\}$, or $\{4, 5, 6\}$ forms a right triangle.

Steps

1. Write as a fraction over 100
   $$0.45 = \frac{45}{100}$$
2. Simplify the fraction by dividing numerator and denominator by 5
   $$\frac{45 \div 5}{100 \div 5} = \frac{9}{20}$$
3. Apply the Pythagorean theorem ($a^2 + b^2 = c^2$) to test the sets
   $$\text{A: } 5^2 + 6^2 = 25 + 36 = 61 \neq 7^2 (49)$$
   $$\text{B: } 6^2 + 8^2 = 36 + 64 = 100 = 10^2 (100)$$
   $$\text{C: } 3^2 + 4^2 = 9 + 16 = 25 \neq 8^2 (64)$$
   $$\text{D: } 4^2 + 5^2 = 16 + 25 = 41 \neq 6^2 (36)$$