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Show that the mid-point of the line segment joining the points $(5, 7)$ and $(3, 9)$ is also the mid-point of the line segment joining the points $(8, 6)$ and $(0, 10)$.
Given:
The line segment joining the points $(5, 7)$ and $(3, 9)$.
The line segment joining the points $(8, 6)$ and $(0, 10)$.
To do:
We have to show that the mid-point of the line segment joining the points $(5, 7)$ and $(3, 9)$ is also the mid-point of the line segment joining the points $(8, 6)$ and $(0, 10)$.
Solution:
Let the mid-point of the line segment joining the points $(5, 7)$ and $(3, 9)$ be $(a,b)$.
Let the mid-point of the line segment joining the points $(8, 6)$ and $(0, 10)$ be $(h,k)$.
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,
\( (a, b)=\left(\frac{5+3}{2}, \frac{7+9}{2}\right) \)
\( \Rightarrow(a, b)=\left(\frac{8}{2}, \frac{16}{2}\right) \)
\( \Rightarrow(a, b)=\left(4, 8\right) \)......(i)
\( (h, k)=\left(\frac{8+0}{2}, \frac{6+10}{2}\right) \)
\( \Rightarrow(h, k)=\left(\frac{8}{2}, \frac{16}{2}\right) \)
\( \Rightarrow(h, k)=\left(4, 8\right) \).......(ii)
From (i) and (ii),
\( (a,b)=(h,k) \)
Hence proved.