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 right-angled triangles: 1) Find the value of e. Step 1: Identify the known sides and the unknown side. This is a right-angled triangle. The hypotenuse is 10, one leg is 7, and the other leg is e. Step 2: Apply the Pythagorean theorem. The Pythagorean theorem states that a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. e^2 + 7^2 = 10^2 Step 3: Solve for e. e^2 + 49 = 100 e^2 = 100 - 49 e^2 = 51 e = sqrt(51) The value of e is sqrt(51). 2) Find the values of s and a. Step 1: Identify the known side, the unknown side, and the unknown angle. This is a right-angled triangle. One leg is 6, the hypotenuse is s, and the angle opposite the leg of length 6 is a. Step 2: Formulate the trigonometric relationship. Using the sine function, () = oppositehypotenuse. (a) = (6)/(s) Step 3: Explain the limitation. With only one side and one angle (both unknown) related by this equation, it is not possible to find unique numerical values for both s and a without additional information (e.g., another side length or another angle). The relationship between them is s = (6)/((a)) or a = ((6)/(s)). 3) Find the length of the unlabeled side. Step 1: Identify the known sides and the unknown side. This is a right-angled triangle. The legs are 7 and 9. The hypotenuse is the unlabeled side. Let's call it h. Step 2: Apply the Pythagorean theorem. 7^2 + 9^2 = h^2 Step 3: Solve for h. 49 + 81 = h^2 130 = h^2 h = sqrt(130) The length of the unlabeled side is sqrt(130). 4) Find the value of x. Step 1: Identify the known sides and the unknown side. This is a right-angled triangle. The hypotenuse is 7, one leg is 3, and the other leg is x. An angle of 37^ is also given. Step 2: Check for consistency. From the perspective of the 37^ angle, the adjacent side is 3 and the hypotenuse is 7. If these values were consistent, (37^) should be equal to (3)/(7). (37^) ≈ 0.7986. (3)/(7) ≈ 0.4286. Since 0.7986 ≠ 0.4286, the given angle of 37^ is inconsistent with the given side lengths of 3 and 7. Step 3: Assume the side lengths are correct and use the Pythagorean theorem to find x. x^2 + 3^2 = 7^2 x^2 + 9 = 49 x^2 = 49 - 9 x^2 = 40 x = sqrt(40) x = sqrt(4 × 10) x = 2sqrt(10) The value of x is 2sqrt(10). That's 2 down. 3 left today — send the next one.