Question 1182583
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The point P has coordinates (a,b). Another point Q is formed by reversing the coordinates of P, 
i.e. Q has coordinates (b,a).
(i) Show that PQ is perpendicular to the line y=x.
(ii) Show that the midpoint, M, of PQ lies on y=x.
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            In this problem, it is assumed that a =/= b --- otherwise, the statement loses sense.

            So, I will assume that a =/= b,  although the problem does not says it explicitly.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Part &nbsp;(&nbsp;i&nbsp;)</U>



<pre>
Let's compare slopes of the line y = x  and the line (segment) PQ.


The line y = x has the slope equal to 1, OBVIOUSLY.


The segment PQ hs the slope  m = {{{increment_y/increment_x}}} = {{{(a-b)/(b-a)}}} = -1.


The slopes 1 and -1 are negatively reciprocal: 1*(-1) = -1; THEREFORE, the line y = x and the segment (the line) PQ are perpendicular.
</pre>


Thus the first statement is proved.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Part &nbsp;(&nbsp;ii&nbsp;)</U>


<pre>
The coordinates of the midpoint are  ({{{(a+b)/2}}},{{{(a+b)/2}}}).


Both x- and y- coordinates are equal.


It proves, that the midpoint lies on the line y = x.
</pre>


Thus the second statement is proved, &nbsp;too.



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The points &nbsp;P &nbsp;and &nbsp;Q &nbsp;are mirror reflections each other about the line &nbsp;&nbsp;y = x.