Question

If the midpoint of a segment joining the points ( 3 , - 1 ) and (m , n ) is ( - 1 , 3 ) then find, ( m , n )​

Answer 1

We're given,

$\longrightarrow \sf{A(3, -1) = ({x}_{1}, {y}_{1})}$

$\longrightarrow \sf{P(-1, 3) = (\alpha, \beta)}$

$\longrightarrow \sf{B(m,n) = ({x}_{2}, {y}_{2})}$

P is the mid-point of line segment AB

$\sf{\\}$

By midpoint formula, we know that if $\sf{P(\alpha, \beta)}$ is the mid-point of line segment AB, where $\sf{A({x}_{1},{y}_{1})}$ and $\sf{B({x}_{2},{y}_{2})}$, then

$\quad \quad \bullet \sf{\alpha = \dfrac{{x}_{1} + {x}_{2}}{2}}$

$\quad \quad \bullet \sf{\beta = \dfrac{{y}_{1} + {y}_{2}}{2}}$

$\sf{\\ \\}$

So, for alpha,

$\longrightarrow \sf{\alpha = \dfrac{{x}_{1} + {x}_{2}}{2}}$

$\longrightarrow \sf{ -1 = \dfrac{3 + m}{2}}$

$\longrightarrow \sf{ -2 = 3 + m}$

$\longrightarrow \underline{\sf{m = -5}}$

$\sf{\\ \\}$

For beta,

$\longrightarrow \sf{\beta = \dfrac{{y}_{1} + {y}_{2}}{2}}$

$\longrightarrow \sf{3 = \dfrac{-1 + n}{2}}$

$\longrightarrow \sf{6 = -1 + n}$

$\longrightarrow \underline{\sf{n = 7}}$

$\sf{\\ \\}$

Hence the required answer is,

$\longrightarrow \underline{\underline{\sf{B = (-5,7)}}}$