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.

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Updated on: 10-Oct-2022

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