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The middle point of the segment of the straight line joining the points $(p, q)$ and $(q,-p)$ is $(r / 2, s / 2)$. What is the length of the segment?
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The correct answer is:
$\left(s^{2}+r^{2}\right)^{1 / 2}$
Two joining points are $(p, q)$ and $(q,-p)$ Mid point of $(p, q)$ and $(q,-p)$ is $\left(\frac{p+q}{2}, \frac{q-p}{2}\right)$
But it is given that the mid-point is $\left(\frac{r}{2}, \frac{s}{2}\right)$.
$\therefore \frac{p+q}{2}=\frac{r}{2}$ and $\frac{q-p}{2}=\frac{s}{2}$
$\Rightarrow p+q=r$ and $q-p=s$
Now, length of segment $=\sqrt{(p-q)^{2}+(q+p)^{2}}$ (by distance formula) $=\sqrt{s^{2}+r^{2}}=\left(s^{2}+r^{2}\right)^{1 / 2}$
But it is given that the mid-point is $\left(\frac{r}{2}, \frac{s}{2}\right)$.
$\therefore \frac{p+q}{2}=\frac{r}{2}$ and $\frac{q-p}{2}=\frac{s}{2}$
$\Rightarrow p+q=r$ and $q-p=s$
Now, length of segment $=\sqrt{(p-q)^{2}+(q+p)^{2}}$ (by distance formula) $=\sqrt{s^{2}+r^{2}}=\left(s^{2}+r^{2}\right)^{1 / 2}$
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