Determine the missing information in the paragraph proof. given: line pq is rotated 90° counterclockwise to form line p’q’. the lines are perpendicular. line pq contains the points (a, b) and (c, d). line p’q’ contains the points (–b, a) and (–d, c). prove: the slopes of perpendicular lines are negative reciprocals. the slopes of lines pq and p’q’ can be determined using the formula the product of these slopes is this product shows that the slopes are negative reciprocals. it is given that the lines are perpendicular and we have shown that the slopes of the lines are negative reciprocals.

The information given is sufficient  for this proof .

Slope of a line passing through   x1 ,y1) and ( x2,y2) is given  by the formula :

            M =   ( y2 - y1)/ ( x2-x1)

Let us start finding the slope of line  PQ  

the  given points are  ( a,b) and ( c,d)  

using the slope formula we get :

slope of line PQ =    m=   ( d-b) /( c-a)

Let us now try finidng slope of the another line  P'Q'  

It is passing through ( -b ,a)  and (-d,c)

using the  formula we get slope of P'Q' = m' =   ( c-a) /( -d -  -b)

                                                                       m'=   ( c-a) /( -d+b)

                                                                        m'=   ( c-a) / -( b-d Let us find the product of  m and m'  :

   ( d-b  )    *     ( c-a)

                         =     -1

(c-a)            - ( b-d)

Because we got product of m and m'  = -1   hence proved   product of perpendicular lines are  negative reciprocal of each other .

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