Step 1: Find the value of $x$ using the Pythagorean theorem. In the right-angled triangle ABC, with the right angle at C, we have: $$AC^2 + BC^2 = AB^2$$ Substitute the given values: $AC = x$, $BC = 8$ cm, $AB = 12$ cm. $$x^2 + 8^2 = 12^2$$ $$x^2 + 64 = 144$$ Subtract 64 from both sides: $$x^2 = 144 - 64$$ $$x^2 = 80$$ Take the square root of both sides: $$x = \sqrt{80}$$ Simplify the square root: $$x = \sqrt{16 \times 5}$$ $$x = 4\sqrt{5}$$ Calculate the numerical value of $x$: $$x \approx 4 \times 2.2360679$$ $$x \approx 8.944 \text{ cm}$$ Rounding to one decimal place, $x \approx 8.9$ cm. Step 2: Find the value of $\theta$ using trigonometry. We can use the sine function, which relates the opposite side to the hypotenuse for angle $\theta$: $$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BC}{AB}$$ Substitute the given values: $$\sin(\theta) = \frac{8}{12}$$ $$\sin(\theta) = \frac{2}{3}$$ To find $\theta$, take the inverse sine (arcsin) of $\frac{2}{3}$: $$\theta = \arcsin\left(\frac{2}{3}\right)$$ $$\theta \approx \arcsin(0.6667)$$ $$\theta \approx 41.81^\circ$$ Rounding to one decimal place, $\theta \approx 41.8^\circ$. Comparing our calculated values with the given options: $x \approx 8.9$ cm and $\theta \approx 41.8^\circ$. This matches option C. The final answer is $\boxed{\text{C. } 8.9\text{cm, } 41.8^\circ}$. Send me the next one 📸