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Difficulty: Very hardProperties of Quadrilaterals

If the diagonals of a convex quadrilateral divide the quadrilateral into four triangles of equal perimeter, then the quadrilateral must be a square. Is this statement true or false?

Answer: Answer

Answer

The statement is false because any non-square rhombus satisfies the condition of being divided into four triangles of equal perimeter, yet it is not a square.
The correct answer is false because the equal-perimeter condition only requires the quadrilateral to be a rhombus (having four equal sides), but does not require the interior angles to be 90 degrees. Any non-square rhombus is a valid counterexample.

Step-by-Step Solution

1
Analyze the given condition of equal perimeters for the four triangles formed by the diagonals of a convex quadrilateral.
The diagonals of any rhombus are perpendicular and bisect each other, dividing the rhombus into four congruent right triangles.
Congruent triangles have identical side lengths, meaning their perimeters are equal. Thus, every rhombus satisfies this property.
2
Determine if all quadrilaterals satisfying this property must be squares.
A square is a regular quadrilateral, meaning it must have both equal side lengths and interior angles of 90 degrees.
To verify if the statement is true, we must test if a non-square rhombus can satisfy the condition.
3
Construct a counterexample using a specific non-square rhombus.
Consider a rhombus with side lengths of 5 units and diagonals of lengths 6 units and 8 units. The diagonals divide it into four right triangles with sides 3, 4, and 5 units. Each triangle has a perimeter of 12 units.
This rhombus has equal perimeters for all four triangles, but its interior angles are not 90 degrees, proving it is not a square.

Key Concept

The relationship between the diagonals and side properties of rhombuses and squares.
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Topics

Properties of Quadrilaterals

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