Please also draw the triangle and explain

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ScanSolve AI Tutor · Accepted 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.

Please also draw the triangle and explain

Answer

Step-by-step solution

Apply the Pythagorean Theorem to ΔADB.

Apply the Pythagorean Theorem to ΔCDB.

Use trigonometry in ΔCDB to relate x to angle C.

Substitute Equation 1 into Equation 2.

Substitute the expression for (b - x) from Step 3 into the equation from Step 4.

Express x in terms of b, a, and cos C.

Substitute the expression for x from Step 6 into the equation from Step 5 and expand.

Simplify the equation and rearrange to solve for c².

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