Question
The middle point of the segment of the straight line joining the points $$\left( {p,\,q} \right)$$ and $$\left( {q,\, - p} \right)$$ is $$\left( {\frac{r}{2},\,\frac{s}{2}} \right).$$ What is the length of the segment ?
A.
$$\frac{{\left[ {{{\left( {{s^2} + {r^2}} \right)}^{\frac{1}{2}}}} \right]}}{2}$$
B.
$$\frac{{\left[ {{{\left( {{s^2} + {r^2}} \right)}^{\frac{1}{2}}}} \right]}}{4}$$
C.
$${\left( {{s^2} + {r^2}} \right)^{\frac{1}{2}}}$$
D.
$$s + r$$
Answer :
$${\left( {{s^2} + {r^2}} \right)^{\frac{1}{2}}}$$
Solution :
Two joining points are $$\left( {p,\,q} \right)$$ and $$\left( {q,\, - p} \right)$$
Mid point of $$\left( {p,\,q} \right)$$ and $$\left( {q,\, - p} \right)$$ is $$\left( {\frac{{p + q}}{2},\,\frac{{q - p}}{2}} \right)$$
But it is given that the mid-point is $$\left( {\frac{r}{2},\,\frac{s}{2}} \right).$$
$$\eqalign{
& \therefore \,\frac{{p + q}}{2} = \frac{r}{2}{\text{ and }}\frac{{q - p}}{2} = \frac{s}{2} \cr
& \Rightarrow p + q = r{\text{ and }}q - p = s \cr} $$
Now, length of segment
$$\eqalign{
& = \sqrt {{{\left( {p - q} \right)}^2} + {{\left( {q + p} \right)}^2}} \,\,\,\,\,\left( {{\text{by distance formula}}} \right) \cr
& = \sqrt {{s^2} + {r^2}} \cr} $$