State whether the statement is true or false.
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, then the quadrilateral is a rectangle.

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True

False

answer is A.

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Detailed Solution

Given that, if diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, then the quadrilateral is a rectangle.
Let the following figure depict the given situation.
Question Image

Δ A O B and Δ C O D

,
AO = OD [radii of same circle]
BO = OC [radii of same circle]

∠ A O B = ∠ C O D

[vertically opposite angles]
By SAS congruence criteria,

Δ A O B ≅ Δ C O D

.
Thus, by C.P.C.T., we get AB = CD. [1]
Similarly, we can prove that

Δ A O D ≅ Δ C O B

and by C.P.C.T. AD = BC. [2]
Since ABCD has opposite sides equal, thus ABCD is a parallelogram.
Now since AC is a diameter,

∠ A D C = 90 °

as the angle in a semicircle is a right angle.
Since a parallelogram with one of its angles as a right angle is a rectangle. Thus, ABCD is a rectangle. Hence, the statement is true.
Therefore, the correct option is 1.

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