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Difficulty: MediumProperties of Quadrilaterals

Determine whether the following statement regarding quadrilateral properties is true or false: Any convex quadrilateral whose diagonals intersect at right angles must be a rhombus.

Answer: Answer

Answer

The statement is false. Perpendicular diagonals alone are not sufficient to prove that a quadrilateral is a rhombus.
The statement is false because perpendicular diagonals are not a sufficient condition to classify a general quadrilateral as a rhombus. Other quadrilaterals, such as kites or general orthodiagonal quadrilaterals with unequal sides, also have diagonals that intersect at right angles.

Step-by-Step Solution

1
Recall the defining properties of a rhombus.
A rhombus is a parallelogram with four congruent sides. Its diagonals are perpendicular bisectors of each other.
To evaluate whether perpendicular diagonals alone guarantee a rhombus.
2
Analyze counterexamples of non-rhombus quadrilaterals with perpendicular diagonals.
A kite has diagonals that intersect at right angles, but only adjacent pairs of sides are congruent, not all four sides. Furthermore, a general quadrilateral can have perpendicular diagonals of arbitrary lengths that do not bisect each other, resulting in four sides of completely different lengths.
A single counterexample disproves a universal mathematical claim.
3
Determine the overall truth value of the statement.
Because perpendicular diagonals are a necessary property of rhombuses but not a sufficient condition for all quadrilaterals, the statement is false.
Concluding the analysis based on geometric counterexamples.

Key Concept

Necessary vs. Sufficient Conditions for Quadrilateral Classification
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Topics

Properties of Quadrilaterals

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