PROBLEM WITH RECTANGLE TRIANGLE

  

                                                                                  

                                                   

 

                                                                   

In a rectangle triangle with sides AB=a,AC=b and BC=c, a perpendicular is drawn from the vertex A to BC, the hypotenuse. Express the length of the perpendicular AD in terms of a,b,c. What is the ratio of BD:DC ?

According to Pythagoras theorem a² + b² = c²

Considering the inner rectangle triangles

AD² + BD² = a² and AD² + CD² = b² . If BD = x and CD = c - x Adding these two relations

c² + 2x² – 2cx + 2AD² = a² + b² = c² or x²- cx + AD² = o Solving this quadratic equation, we get x = [c – (c²-4AD²]^1/2 / 2.

 On subtraction, CD² – BD² = c²- 2cx =a² + b² -2cx = b² – a². Or a² = cx or x = BD = a²/c and c-x = CD =

 b² /c. Substituting this value of x in the above equation we get,

AD = ab/c and BD:CD = a²: b²