Asked by No name
Which statement can be used to prove that a given parallelogram is a rectangle?
Responses
The opposite sides of the parallelogram are congruent.
The opposite sides of the parallelogram are congruent.
The diagonals of the parallelogram are congruent.
The diagonals of the parallelogram are congruent.
The diagonals of the parallelogram bisect the angles.
The diagonals of the parallelogram bisect the angles.
The opposite angles of the parallelogram are congruent.
All Answers 1
Answered by
GPT-5 mini
AI
The diagonals of the parallelogram are congruent.
Reason: Let adjacent side vectors be u and v. The diagonals are u+v and u−v; if their lengths are equal then |u+v|^2 = |u−v|^2, so
(u+v)·(u+v) = (u−v)·(u−v) ⇒ 4(u·v) = 0 ⇒ u·v = 0.
Thus adjacent sides are perpendicular, so the parallelogram has right angles and is a rectangle.
Reason: Let adjacent side vectors be u and v. The diagonals are u+v and u−v; if their lengths are equal then |u+v|^2 = |u−v|^2, so
(u+v)·(u+v) = (u−v)·(u−v) ⇒ 4(u·v) = 0 ⇒ u·v = 0.
Thus adjacent sides are perpendicular, so the parallelogram has right angles and is a rectangle.
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