Question

find the fourth vertix of the parallelogram whose consecutive vertices are(8,4),(5,7),(-1,1)​

Answer 1

Let the vertices be,

• A(8,4)
• B(5,7)
• C(-1,1)
• D(x,y)

In a parallelogram, diagonals bisect each other.

•°• Midpoint of AC = Midpoint of BD

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We know that, Midpoint formula =

$( \frac{ {x}_{1} + {x}_{2} }{2} ,\frac{ {y}_{1} + {y}_{2} }{2})$

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Now,

$\rightarrow \: ( \frac{ {8 + ( - 1)}}{2} , \frac{ {4 + 1}}{2} ) = ( \frac{ {5 + x}}{2} , \frac{ {7 + y}}{2} ) \\ \rightarrow \: ( \frac{ {8 - 1}}{2} , \frac{ {5}}{2} ) = ( \frac{ {5 + x }}{2} , \frac{ {7 + y}}{2} ) \\ \rightarrow \: ( \frac{ {7}}{2} , \frac{ {5}}{2} ) = ( \frac{ {5 + x }}{2} , \frac{ {7 + y}}{2} ) \\ \underline{equating \: x \: and \: y \: coordinates \colon} \\ \rightarrow \frac{7}{2 } = \frac{5 + x}{2} \\ \rightarrow \: 7 = 5 + x \\ \rightarrow \: 7 - 5 = x \\ \rightarrow \: 2 = x \\ \therefore \: x \: = 2 \\ \underline{also \colon} \\ \rightarrow \: \frac{5}{2} = \frac{7 + y}{2} \\ \rightarrow \: 5 = 7 + y \\ \rightarrow \: 5 - 7 = y \\ \rightarrow - 2 = y \\ \therefore \: y \: = - 2$

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(x,y) = (2,-2)

•°• The fourth vertice is, (2,-2).

Answer 2

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