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 make an accurate drawing of the triangle and find the length d: Instructions for drawing the triangle: 1. Draw a line segment of length 8.9 cm. This will be the base of your triangle. 2. At one end of the 8.9 cm line segment, place the center of a protractor. Measure and mark an angle of 78^. 3. From the vertex where you measured 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 the length d. 5. Measure the length of side d with a ruler and round it to 1 decimal place. Calculation of length d using the Law of Cosines: Since we are given two sides and the included angle (SAS), we can use the Law of Cosines: d^2 = a^2 + b^2 - 2ab () Here, a = 7.5 cm, b = 8.9 cm, and = 78^. Step 1: Substitute the 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^) ≈ 0.20791169 Step 5: Substitute these values back into the equation for d^2. d^2 = 56.25 + 79.21 - 133.5 × 0.20791169 d^2 = 135.46 - 27.750259 d^2 = 107.709741 Step 6: Take the square root to find d. d = sqrt(107.709741) d ≈ 10.37833 Step 7: Round the length d to 1 decimal place. d ≈ 10.4 cm The measured length d in your drawing should be approximately 10.4 cm. d = 10.4 cm