2007 AMC 10A Problem 14

1. It is a right triangle. $3^2 + 4^2 = 9 + 16 = 25 = 5^2$, so the sides in ratio $3 : 4 : 5$ form a right triangle, and the "$5$" side is the longest side (the hypotenuse).
2. The hypotenuse is a diameter. When a right triangle sits inside a circle with all three corners on the circle, its hypotenuse always goes straight through the center. So the hypotenuse equals the diameter, $2 \times 3 = 6$.
3. Find the sides. The "$5$" part is $6$, so each part is $\frac{6}{5} = 1.2$. The sides are
$$3 \times 1.2 = 3.6, \qquad 4 \times 1.2 = 4.8, \qquad 5 \times 1.2 = 6.$$
4. Area. The two shorter sides meet at the right angle, so they are the base and height:
$$\frac12 \times 3.6 \times 4.8 = 1.8 \times 4.8 = 8.64.$$

The answer is $\boxed{\textbf{(A)}\ 8.64}$.

Two facts do all the work. First, $3 : 4 : 5$ means a right triangle. Second, a right triangle drawn inside a circle has its hypotenuse as a diameter. Once you know one real side length, the ratio gives the rest.

Scale the area directly. The plain $3$-$4$-$5$ triangle has area $\frac12 \times 3 \times 4 = 6$. Every side is multiplied by $\frac65$, so the area is multiplied by $\left(\frac65\right)^2 = \frac{36}{25}$. That gives $6 \times \frac{36}{25} = \frac{216}{25} = 8.64$.