Answers

The Brainliest Answer!
2015-04-22T22:26:36+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
Let  A (a, b)  be the vertex containing the right angle.
Let  the equation of hypotenuse BC be  y = m x + c
we are given  a, b, m, and c.

Let equation of AB be :  y = p x + q  and   point A is on AB.
                 b = a p + q      =>  q = b - a p 

    So equation of AB :    y = p (x - a) + b  ,    p is a parameter that is the slope of AB.  There are infinitely many  lines AB possible.

    equation of AC :  y = -1/p ( x - a) + b ,  as  A(a, b) on AC and its slope is -1/p as AC is perpendicular to AB.

 Now, the point B will be the intersection of BC : y = m x + c and AB :
       y = m x + c = p(x - a) + b
               x (m -p) = b - c - a p
               x = (b - c- a p) / (m - p)
               y = m (b - c - a p) / (m - p)

  Now, the point of intersection of BC and AC is C:
         y = m x + c = -1/p * (x - a) + b
             x (m + 1/ p) = b - c + a/p
             x = (p b - p c + a) / (pm + 1)
             y = m (p b - p c + a) / (p m + 1)
================
The solution: 

   Given hypotenuse BC:  y = m x + c  and  A (a, b) , the vertex with right angle,  we find the equations of AB, AC and the coordinates of B and C with a parameter p.  There are infinitely many such triangles.

   AB:   y = p (x - a) + b
   AC :  y = -1/p (x - a) + b
   B = [ (b - c- a p) / (m - p),    m (b - c - a p) / (m - p)  ]
   C =  [ (p b - p c + a) / (pm + 1) ,    m (p b - p c + a) / (p m + 1)  ]

2 5 2
oh god ! ok thanks sir
Length of hypotenuse will be:
BC^2 = [ (pm+1) (b-c-ap) - (m-p)(pb-pc+a) ]^2 + [ m (pm+1)(b-c-ap) - m (m-p)(pb - pc +a) ]^2
= (1+m^2) [ pmb - pmc - ap^2m + b - c - ap - mpb + mpc - ma + p^2b - p^2c+pa ]^2
= (1+m^2) [ - am p^2 + b - c - ma + b p^2 - c p^2 ]^2
BC^2 = (1+m^2) (1+ p^2) (b - c - a m)
So length of the hypotenuse of the triangle ABC depends on parameter p, with minimum value when p = 0.
length of side AB² = [ (b - am)² (1 + p²) - 2 c (b - am) (1 + pm) + c² (1+ m²) ] / (m-p)²
OK