Question

Find vertex M of parallelogram LMNO, where L-(10,-5), N=(-4,-2) and 0(-14, -2).​

Answer 1

$\large\underline\blue{\bold{Given :-  }}$

$\begin{gathered}\begin{gathered}\sf \: A \: ||gm \: LMNO \: with \: vertex \begin{cases} &\sf{L(10,-5)} \\ &\sf{N(-4,-2)}\\ &\sf{0(-14, -2).} \end{cases}\end{gathered}\end{gathered}$

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$\large\underline\blue{\bold{To \: Find :-  }}$

$\begin{gathered}\begin{gathered}\bf Find \begin{cases} &\sf{fourth \: vertex \: of \: parallelogram \: M} \end{cases}\end{gathered}\end{gathered}$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

❥︎Let us consider a line segment joining the points A and B

$\sf \:  ⟼A(x_1,y_1) \: and \: B(x_2,y_2)$

and Let C (x, y) be the midpoint of AB, then coordinates of C is given by

$\bf \:( x, y) = (\dfrac{x_1+x_2}{2} , \dfrac{y_1+y_2}{2} )$

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$\large\underline\purple{\bold{Solution :- }}$

Let the coordinates of fourth vertex be M (x, y).

❥︎ We know,

❥︎ In a parallelogram, diagonals bisect each other.

❥︎ So, in parallelogram LMNO, LN and MO are diagonals.

❥︎ Hence, midpoint of LN = midpoint of MO

$\sf \:  ⟼\bf \:(\dfrac{10 - 4}{2} , \dfrac{ - 5 - 2}{2} ) = (\dfrac{x - 14}{2} , \dfrac{y - 2}{2} )$

$\sf \:  ⟼(\dfrac{6}{2} , \dfrac{ - 7}{2} ) = (\dfrac{x - 14}{2} , \dfrac{y - 2}{2} )$

❥︎ On comparing, we get

$\sf \:  ⟼x - 14 = 6 \: and \: y - 2 = - 7$

$\bf\implies \:x = 20 \: and \: y \: = - 5$

❥︎ Hence, the coordinates of M is (20, -5)

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${\boxed{\boxed{\bf{Hence, \: the \: coordinates \: of \: M \: is \: (20, -5)</p><p>}}}}$

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