Question

claudia knows the given information about quadrilateral fghi.

• not all of the sides are congruent.
• diagonals \\(\overline{fh}\\) and \\(\overline{gi}\\) are congruent.

claudia claims that fghi must be a rectangle because rectangles are parallelograms with congruent
what additional piece of information is necessary for claudia’s claim to be true?
ⓐ the diagonals bisect each other.
ⓑ the diagonals are perpendicular.
ⓒ the diagonals are angle bisectors.
ⓓ the diagonals intersect at a point p such that \\(\frac{fp}{hp} = \frac{gp}{ip}\\).

Explanation:

To determine the additional information needed for Claudia's claim (that \( FGHI \) is a rectangle) to be true, we analyze the properties of rectangles and parallelograms:

1. A rectangle is a parallelogram with congruent diagonals. For \( FGHI \) to be a parallelogram (a prerequisite for being a rectangle), its diagonals must bisect each other (a defining property of parallelograms).
2. Analyzing the options:

• Option A: If diagonals bisect each other, \( FGHI \) is a parallelogram. Combined with congruent diagonals (given), this makes it a rectangle (since a parallelogram with congruent diagonals is a rectangle).
• Option B: Diagonals being perpendicular is a property of rhombuses (or squares), not necessary for a rectangle.
• Option C: Diagonals being angle bisectors is not a property of rectangles (except squares, but the problem states not all sides are congruent, so not a square).
• Option D: The ratio condition describes similar triangles but does not guarantee \( FGHI \) is a parallelogram.

Answer:

A. The diagonals bisect each other.