1. **Problem Statement:** Express the cosine of angle $K$ in simplest fractional form given a right triangle with sides opposite vertices $J$, $K$, and $L$ measuring 65, 52, and 39 respectively, and the right angle at vertex $J$. 2. **Recall the definition of cosine in a right triangle:** $$\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$ 3. **Identify the sides relative to angle $K$:** - The side opposite $K$ is 52. - The hypotenuse is the side opposite the right angle $J$, which is 65. - The adjacent side to angle $K$ is the side opposite $L$, which is 39. 4. **Calculate $\cos K$ using the adjacent and hypotenuse sides:** $$\cos K = \frac{39}{65}$$ 5. **Simplify the fraction:** $$\frac{39}{65} = \frac{\cancel{13} \times 3}{\cancel{13} \times 5} = \frac{3}{5}$$ 6. **Final answer:** $$\cos K = \frac{3}{5}$$ This means the cosine of angle $K$ is $\frac{3}{5}$ in simplest terms.