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
Here are the solutions for the marked sides: a) Step 1: Identify the trigonometric relationship. In the right-angled triangle, we are given the angle 45^, the side opposite to it (80), and we need to find the hypotenuse (x). The sine function relates these: () = oppositehypotenuse Step 2: Substitute the values and solve for x. (45^) = (80)/(x) x = (80)/((45^)) Since (45^) = sqrt(2)2: x = (80)/(sqrt(2))2 = (160)/(sqrt(2)) To rationalize the denominator, multiply the numerator and denominator by sqrt(2): x = 160sqrt(2)2 = 80sqrt(2) The value of x is 80sqrt(2). b) This problem involves two right-angled triangles sharing a common altitude. Let h be the common altitude. Step 1: Calculate the altitude h using the left triangle. In the left right-angled triangle, the hypotenuse is 8 cm and the angle is 60^. (60^) = (h)/(8 cm) h = 8 cm × (60^) = 8 cm × sqrt(3)2 = 4sqrt(3) cm Step 2: Calculate the hypotenuse Hy of the right triangle. In the right right-angled triangle, the altitude is h = 4sqrt(3) cm and the angle is 45^. (45^) = (h)/(Hy)