1. **Problem statement:** We are given a right triangle with vertices Q, P, and R, where the right angle is at Q. The hypotenuse PR has length $\sqrt{89}$, and the angle at vertex R is $28^\circ$. We need to find the length of side QR. 2. **Relevant formula:** In a right triangle, the side opposite an angle can be found using the sine function: $$\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$$ Here, $\theta = 28^\circ$, the hypotenuse is $\sqrt{89}$, and the opposite side to angle R is QR. 3. **Set up the equation:** $$\sin(28^\circ) = \frac{QR}{\sqrt{89}}$$ 4. **Solve for QR:** $$QR = \sqrt{89} \times \sin(28^\circ)$$ 5. **Calculate the value:** First, approximate $\sqrt{89} \approx 9.433$. Next, calculate $\sin(28^\circ) \approx 0.4695$. 6. **Multiply:** $$QR \approx 9.433 \times 0.4695 = 4.429$$ 7. **Round to the nearest tenth:** $$QR \approx 4.4$$ **Final answer:** $$\boxed{4.4}$$