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In quadrilateral PQRS, the vertices are P(1, 3), Q(2, 6), R(5, 5), and S(6, 2). Prove that PQRS is a parallelogram using the vector method.

Answer :

Final answer:

By using vector calculations, we have shown that opposite sides of the quadrilateral PQRS are both parallel and equal in magnitude, which proves that PQRS is a parallelogram according to the principles of affine geometry.

Explanation:

To prove that the quadrilateral PQRS with vertices P(1,3), Q(2,6), R(5,5), and S(6,2) is a parallelogram using vector methods, we need to demonstrate that opposite sides are parallel and equal in length. This can be done by comparing the vectors that represent these sides.

First, calculate the vectors PQ and RS:

PQ = Q - P

= (2 - 1, 6 - 3)

= (1, 3)

RS = S - R

= (6 - 5, 2 - 5)

= (1, -3)

Similarly, calculate the vectors QR and PS:

QR = R - Q

= (5 - 2, 5 - 6)

= (3, -1)

PS = S - P

= (6 - 1, 2 - 3)

= (5, -1)

Now examine PQ and RS. They are not only parallel (as shown by having the same direction ratios) but also equal in magnitude but opposite in direction, indicating that they are of the same length but on opposite sides of the quadrilateral.

Similarly, QR and PS are also parallel (having proportional direction ratios) and equal in magnitude. This proves that both pairs of opposite sides of quadrilateral PQRS are parallel and equal, satisfying the definition of a parallelogram.

Therefore, by the vector method, we have proved that PQRS is a parallelogram according to A1 Constructibility of parallelograms and A2 Symmetric treatment of the sides of a parallelogram. It satisfies the conditions mentioned in the axioms, with side PQ parallel to side RS and side QR parallel to side PS.

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