midpoint
- Thread starter humakhan
- Start date
pka
Elite Member
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- Jan 29, 2005
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Given the points \(\displaystyle P(a,b)\quad \& \quad Q(c,d)\) the midpoint of \(\displaystyle \overline {PQ}\) is the average of the coordinates: \(\displaystyle M\left( {\frac{{a + c}}{2},\frac{{b + d}}{2}} \right)\)
The slope of \(\displaystyle \overline {PQ}\) is \(\displaystyle \frac{{d - b}}{{c - a}}\).
The slope of \(\displaystyle \overline {PM}\) is \(\displaystyle \frac{{\frac{{d + b}}{2} - b}}{{\frac{{a + c}}{2} - a}} = \frac{{d - b}}{{c - a}}\) the same slope!
That proves that the midpoint must be on the line segement!
The slope of \(\displaystyle \overline {PQ}\) is \(\displaystyle \frac{{d - b}}{{c - a}}\).
The slope of \(\displaystyle \overline {PM}\) is \(\displaystyle \frac{{\frac{{d + b}}{2} - b}}{{\frac{{a + c}}{2} - a}} = \frac{{d - b}}{{c - a}}\) the same slope!
That proves that the midpoint must be on the line segement!
