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Difficulty: MediumProperties of Quadrilaterals

Determine whether the following statement is true or false:

The quadrilateral formed by connecting the midpoints of the four consecutive sides of any rhombus is always a rectangle.

Answer: Answer

Answer

The statement is true because the diagonals of a rhombus are perpendicular, which forces the adjacent sides of the midpoint quadrilateral (which are parallel to the diagonals) to meet at right angles.
The statement is true because the diagonals of any rhombus are perpendicular. By the Midsegment Theorem, the sides of the quadrilateral formed by connecting adjacent midpoints are parallel to these diagonals, ensuring all four interior angles are 90∘90^\circ, which satisfies the definition of a rectangle.

Step-by-Step Solution

1
Apply the Triangle Midsegment Theorem to rhombus ABCDABCD with diagonals ACAC and BDBD.
Let P,Q,R,P, Q, R, and SS be the midpoints of sides AB,BC,CD,AB, BC, CD, and DADA respectively. Segment PQPQ is parallel to diagonal ACAC and has length 12AC\frac{1}{2}AC. Segment QRQR is parallel to diagonal BDBD and has length 12BD\frac{1}{2}BD.
A line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
2
Use the geometric properties of a rhombus's diagonals.
In any rhombus, the diagonals ACAC and BDBD are perpendicular to each other (AC⊥BDAC \perp BD).
Perpendicular diagonals are a key defining property of all rhombuses.
3
Determine the interior angle measures of quadrilateral PQRSPQRS.
Because PQ∥ACPQ \parallel AC and QR∥BDQR \parallel BD, and AC⊥BDAC \perp BD, the sides PQPQ and QRQR must be perpendicular (PQ⊥QRPQ \perp QR). Therefore, all four interior angles of quadrilateral PQRSPQRS are 90∘90^\circ.
If two lines are parallel to two mutually perpendicular lines, they are also mutually perpendicular.
4
Classify quadrilateral PQRSPQRS.
Quadrilateral PQRSPQRS has four right angles, so it is by definition a rectangle.
Any quadrilateral with four right angles is a rectangle.

Key Concept

Properties of Rhombus Diagonals and Midpoint Quadrilaterals
Estimated Time:1m 0s
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Topics

Properties of Quadrilaterals

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