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
To find the length d, we will use the Law of Cosines, as we are given two sides and the included angle (SAS). The Law of Cosines states: d^2 = a^2 + b^2 - 2ab () where a = 7.5 cm, b = 8.9 cm, and = 78^. Step 1: Substitute the given values into the formula. d^2 = (7.5)^2 + (8.9)^2 - 2(7.5)(8.9) (78^) Step 2: Calculate the squares of the sides. (7.5)^2 = 56.25 (8.9)^2 = 79.21 Step 3: Calculate the product 2ab. 2(7.5)(8.9) = 15 × 8.9 = 133.5 Step 4: Find the value of (78^). (78^) ≈ 0.20791 Step 5: Substitute these values back into the equation for d^2. d^2 = 56.25 + 79.21 - 133.5 × 0.20791 d^2 = 135.46 - 27.750285 d^2 = 107.709715 Step 6: Take the square root to find d. d = sqrt(107.709715) d ≈ 10.3783 Step 7: Round the length d to 1 decimal place. d ≈ 10.4 cm To make an accurate drawing of the triangle: 1. Draw a line segment of length 8.9 cm. 2. At one end of this line segment, use a protractor to measure and mark an angle of 78^. 3. From the vertex of the 78^ angle, draw a line segment of length 7.5 cm along the 78^ mark. 4. Connect the end of the 8.9 cm line segment to the end of the 7.5 cm line segment. This third side is d. 5. Measure the length of side d with a ruler. It should be approximately 10.4 cm. The measured length d is: d = 10.4 cm