Complete the proof of the Law of Cosines for the case when ABC is an acute
Chapter 8, Problem 57(choose chapter or problem)
Complete the proof of the Law of Cosines for the case when ABC is an acute triangle.Given: ABC is acute with side lengths a, b, and c.Prove: a 2 = b 2 + c 2 - 2bc cos AProof: Draw the altitude from C to AB . Let h be the length of this altitude.It divides AB into segments of lengths x and y. By the Pythagorean Theorem,a 2 = a. ? , and b. ? = h 2 + x 2 . Substitute y = c - x into the first equationto get c. ? . Rearrange the terms to get a 2 = (h 2 + x 2) + c 2 - 2cx. Substitute theexpression for b 2 to get d. ? . From the diagram, cos A =__xb . So x = e. ? .Therefore a 2 = b 2 + c 2 - 2bc cos A by f. ? .
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