This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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 × 5) x = 4sqrt(5) Calculate the numerical value of x: x ≈ 4 × 2.2360679 x ≈ 8.944 cm Rounding to one decimal place, x ≈ 8.9 cm. Step 2: Find the value of using trigonometry. We can use the sine function, which relates the opposite side to the hypotenuse for angle : () = OppositeHypotenuse = (BC)/(AB) Substitute the given values: () = (8)/(12) () = (2)/(3) To find , take the inverse sine (arcsin) of (2)/(3): = ((2)/(3)) ≈ (0.6667) ≈ 41.81^ Rounding to one decimal place, ≈ 41.8^. Comparing our calculated values with the given options: x ≈ 8.9 cm and ≈ 41.8^. This matches option C. The final answer is C. 8.9cm, 41.8^. Send me the next one 📸