The side lengths for delta ABC are a b and c The height of t

The side lengths for delta ABC are a, b, and c The height of the triangle is h, and CD = x. Complete the steps below to prove the Law of Cosines. When filling in the blanks, you may use the letters a, b, c, x, and h Use the Pythagorean Theorem to find a^2 a^2 = (c - x)^2 + h^2 a^2 = (b - x)^2 + h^2 a^2 = b^2 + h^2 a^2 = x^2 + h^2 Use the Pythagorean theorem to find c^2. C^2 = (b - x)^2 + h^2 c^2 = x^2 + h^2 c^2 = (x + b)^2 + h^2 c^2 = (a - x)^2 + h^2 Use the answer from Part 2 to fill in the blanks. C^2 = x^2 + h^2 + ^2 -2.. Use the answers from Parts 1 and 3 to fill in the blanks. C^2 = ^2 + ^2 -2.. Use trigonometry to fill in the blank. X = cos C Use the answers from Parts 4 and 5 to fill in the blanks. C^2 = ^2 + ^2 - 2.. cos C

Solution

1) Traingle BDC: pythagoras theorem:

a^2 = x^2 + h^2

2) Triangle BDA: apply pythagoras theorem:

c^2 =h^2 +(b -x)^2

3) c^2 = h^2 +b^2 +x^2 -2b*x

4) from part 1 ) a^2 = x^2 + h^2

So, c^2 = b^2 + a^2 -2b*x

5) Traingle BDC : cosC = x/a

S0, x = a*cosC

6)  c^2 = b^2 + a^2 -2b*acosC