Here's Why It Works

In this Δ ABC cap delta with altitude h, let A D = eh d equals x.

Then D B = c − x . d b equals c minus x .

In Δ ADC , cap delta

b 2 = x 2 + h 2 b squared , equals , x squared , plus , h squared and

cos A = x b cosine eh equals , x over b or x = b cos A . x equals b cosine eh .

In Δ CBD , cap delta

a 2 = ( c − x ) 2 + h 2 Pythagorean Theorem   = c 2 − 2 c x + x 2 + h 2 Square the binomial .   = c 2 − 2 c x + b 2 Substitute  b 2  for  x 2 + h 2 .   = c 2 − 2 c b cos A + b 2 Substitute  b  cos  A  for  x .   = b 2 + c 2 − 2 b c cos A Commutative Properties of Addition and Multiplication table with 5 rows and 3 columns , row1 column 1 , eh squared , column 2 equals . open , c minus x , close squared . plus , h squared , column 3 cap pythagoreancap theorem , row2 column 1 , , column 2 equals , c squared , minus 2 c x plus , x squared , plus , h squared , column 3 cap squarethebinomial . . , row3 column 1 , , column 2 equals , c squared , minus 2 c x plus , b squared , column 3 cap substitute . b squared , for , x squared , plus , h squared , . , row4 column 1 , , column 2 equals , c squared , minus 2 c b cosine eh plus , b squared , column 3 cap substitute . b , cos eh , for , x . , row5 column 1 , , column 2 equals , b squared , plus , c squared , minus 2 b c cosine eh , column 3 cap commutativecap propertiesofcap additionandcap multiplication , end table

The last equation is the Law of Cosines.

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