High School

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The diagonals of quadrilateral WXYZ intersect at R. Given that R is the midpoint of WY and XZ, which additional statement ensures that WXYZ is a square?

A. WY ≅ XZ
B. WY ⊥ XZ
C. ∠WZY = 90°
D. WX ≅ YZ

Answer :

Final answer:

Option (B) WY⊥XZ, which indicates that the diagonals are perpendicular, is the additional statement that ensures quadrilateral WXYZ is a square, as it leads to the conclusion that all sides and angles meet the criteria for a square.

Explanation:

The question asks which additional statement ensures that quadrilateral WXYZ, where R is the midpoint of diagonals WY and XZ, is a square. To ascertain that WXYZ is a square, several characteristics must be assessed, including the lengths of the sides and the angles involved.

The option (B) WY⊥XZ indicates that the diagonals are perpendicular to each other. This fact, combined with the given that R is the midpoint of both diagonals, implies that all four angles formed at point R are right angles. Since both diagonals are equal in length (due to R being the midpoint), and they bisect each other at right angles, we can deduce that all four sides of the quadrilateral must be of equal length. Thus, the quadrilateral must have four right angles and four sides of equal length, which are the defining characteristics of a square.

Therefore, option (B) WY⊥XZ is the correct answer.

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