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 prove the Cosine Rule c² = a² + b² - 2ab cos C for an acute-angled triangle ΔABC, we can use the Pythagorean theorem and basic trigonometry. Here's how to set up the triangle and the proof: Imagine the triangle: Consider an acute-angled triangle ABC. Label the vertices A, B, C. Label the side opposite vertex A as a (side BC). Label the side opposite vertex B as b (side AC). Label the side opposite vertex C as c (side AB). Draw an altitude (a perpendicular line) from vertex B to side AC. Label the point where this altitude meets AC as D. This altitude BD divides the triangle ABC into two right-angled triangles: ΔADB and ΔCDB. Let the length of the altitude BD be h. Let the length of the segment AD be x. Then the length of the segment DC will be b - x (since AC = b). Step 1: Apply the Pythagorean Theorem to ΔADB. In the right-angled triangle ΔADB: AB² = AD² + BD² c² = x² + h² This gives us our first equation: h² = c² - x² (Equation 1) Step 2: Apply the Pythagorean Theorem to ΔCDB. In the right-angled triangle ΔCDB: BC² = DC² + BD² a² = (b - x)² + h² (Equation 2) Step 3: Use trigonometry in ΔCDB to relate x to angle C. In the right-angled triangle ΔCDB: cos C = (adjacent side)/(hypotenuse) cos C = DC / BC cos C = (b - x) / a Rearrange this to express (b - x): b - x = a cos C Step 4: Substitute Equation 1 into Equation 2. Substitute h² = c² - x² from Equation 1 into Equation 2: a² = (b - x)² + (c² - x²) Step 5: Substitute the expression for (b - x) from Step 3 into the equation from Step 4. We have b - x = a cos C. So, a² = (a cos C)² + c² - x² Step 6: Express x in terms of b, a, and cos C. From b - x = a cos C (from Step 3), we can write: x = b - a cos C Step 7: Substitute the expression for x from Step 6 into the equation from Step 5 and expand. a² = (a cos C)² + c² - (b - a cos C)² a² = a² cos² C + c² - (b² - 2ab cos C + a² cos² C) a² = a² cos² C + c² - b² + 2ab cos C - a² cos² C Step 8: Simplify the equation and rearrange to solve for c². The terms a² cos² C and -a² cos² C cancel out: a² = c² - b² + 2ab cos C Rearrange the terms to isolate c²: c² = a² + b² - 2ab cos C This proves the Cosine Rule for an acute-angled triangle. What's next? Send 'em 📸