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Find the distance of the point $(1, 2)$ from the mid-point of the line segment joining the points $(6, 8)$ and $(2, 4)$.
Given:
The line segment joining the points $(6, 8)$ and $(2, 4)$.
To do:
We have to find the distance of the point $(1, 2)$ from the mid-point of the line segment joining the points $(6, 8)$ and $(2, 4)$.
Solution:
Let the mid-point of the line segment joining the points $(6, 8)$ and $(2, 4)$ be $(a,b)$.
We know that,
Mid-point of a line segment joining points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1+y_1}{2}, \frac{y_1+y_2}{2})$
Therefore,
\( (a, b)=\left(\frac{6+2}{2}, \frac{8+4}{2}\right) \)
\( \Rightarrow(a, b)=\left(\frac{8}{2}, \frac{12}{2}\right) \)
\( \Rightarrow(a, b)=\left(4, 6\right) \)
The distance between two points \( \mathrm{A}\left(x_{1}, y_{1}\right) \) and \( \mathrm{B}\left(x_{2}, y_{2}\right) \) is \( \sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \).
Therefore,
The distance between \( (1,2) \) and \( (4,6) =\sqrt{(4-1)^2+(6-2)^2} \)
\( =\sqrt{3^2+4^2} \)
\( =\sqrt{9+16} \)
\( =\sqrt{25} \)
\( =5 \) units
The distance of the point $(1, 2)$ from the mid-point of the line segment joining the points $(6, 8)$ and $(2, 4)$ is 5 units.
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