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Prove using vectors that a quadrilateral is a rectangle if and only if its diagonals are congruent and bisect each other. a) A quadrilateral is a rectangle if its diagonals are not congruent. b) A quadrilateral is not a rectangle even if its diagonals bisect each other. c) A quadrilateral is a rectangle if its diagonals are congruent and bisect each other. d) A quadrilateral is a rectangle if its diagonals are congruent but do not bisect each other.

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Let's dissect each choice separately:

a) A quadrilateral is a rectangle if its diagonals are not congruent:

This statement is incorrect. A rectangle is a type of quadrilateral where all angles are right angles, and its diagonals are congruent (equal in length), which means they must be the same length. If a quadrilateral doesn't have congruent diagonals, it won't satisfy the rectangle's properties hence, it cannot be a rectangle.

b) A quadrilateral is not a rectangle even if its diagonals bisect each other:

Generally, this statement is accurate. Though a rectangle's diagonals bisect each other, not every quadrilateral does. Furthermore, a quadrilateral whose diagonals bisect each other may not necessarily be a rectangle—it could also be a rhombus, for instance. Hence, an accurate conclusion can't be made solely from this characteristic.

c) A quadrilateral is a rectangle if its diagonals are congruent and bisect each other:

This statement seems accurate. A rectangle has congruent (equal length) diagonals that bisect each other, meaning they intersect each other at their midpoints. Hence, a quadrilateral that displays these same characteristics is most likely a rectangle.

d) A quadrilateral is a rectangle if its diagonals are congruent but do not bisect each other:

This statement is false. A rectangle's diagonals are not only congruent but also bisect each other, intersecting each other at their midpoint. If a quadrilateral's diagonals are congruent but do not bisect each other, it cannot meet the criteria to be a rectangle.

In conclusion, from all the provided statements, only option c) correctly describes the properties of a rectangle. That is, a quadrilateral is a rectangle if and only if its diagonals are congruent and bisect each other. Hence, the answer is 'C'.

Answer: c) A quadrilateral is a rectangle if its diagonals are congruent and bisect each other.

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