Question

Determine the missing information in the paragraph proof.

Answer 1

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)

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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 .

                 

Answer 2

Answer: The product of the slopes is -1

Step-by-step explanation:

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).

The slope of  PQ is given by :-

$\text{slope}_1=\frac{d-b}{c-a}$

The slope of  P'Q' is given by :-

$\text{slope}_2=\frac{c-a}{-d-(-b)}=\frac{c-a}{-d+b}=\frac{c-a}{b-d}$

The product of the slopes of PQ and P'Q'  is given by:-

$\text{slope}_1\times \text{slope}_2=\frac{d-b}{c-a}\times\frac{c-a}{b-d}\\\\=\frac{d-b}{-(d-b)}\\\\=-1$

Hence, the product of the slopes is -1 .