Three vertices of a parallelogram are ( a+ b, a -b), (2a+b, 2a- b), (a - b, a + b). Find the fourth vertex.

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Solution

Given three vertices of a parallelogram are A( a+ b, a -b), B (2a+b, 2a- b), C(a - b, a + b).

As we know the vertices of the paralllelogram bisect each other so let's consider the bisection point of the diagonals be O (X,Y).

To find the mid point of point A and C;

Apply midpoint formula for finding O

$X equals fraction numerator open parentheses a plus b close parentheses plus open parentheses a minus b close parentheses over denominator 2 end fraction equals fraction numerator 2 a over denominator 2 end fraction equals a Y equals fraction numerator open parentheses a minus b close parentheses plus open parentheses a plus b close parentheses over denominator 2 end fraction equals fraction numerator 2 a over denominator 2 end fraction equals a h e n c e space t h e space p o i n t space O space equals left parenthesis a comma a right parenthesis$
Let the fourth point be D(p,q)

now applying section formula for points Band D
$X equals a equals fraction numerator open parentheses 2 a plus b close parentheses plus open parentheses p close parentheses over denominator 2 end fraction 2 a equals 2 a plus b plus p p plus b equals 0 p equals negative b f o r space y space c o o r d i n a t e s Y equals a equals fraction numerator open parentheses 2 a minus b close parentheses plus q over denominator 2 end fraction 2 a equals 2 a minus b plus q q minus b equals 0 q equals b s o space t h e space c o o r d i n a t e s space o f space p o i n t space D left parenthesis p comma q right parenthesis equals left parenthesis negative b comma b right parenthesis$

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