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# If the diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, then the kind of the quadrilateral is:(a) a parallelogram with no angle equal to a right angle(b) a rectangle(c) a trapezium(d) a scalene quadrilateral

Last updated date: 16th Sep 2024
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Hint: Here, we need to find the kind of the given cyclic quadrilateral. We will use the property of an angle in a semi-circle to find the measures of the interior angles of the cyclic quadrilateral. Using the measures of the angles, we can determine the type of the quadrilateral.

First, we will draw the required figure for the problem.

Here, O is the centre of the circle, AC and BD are the diameters of the circle, and the diagonals of the quadrilateral ABCD. ABCD is a cyclic quadrilateral.
Now, we can observe that the arc ADC and the diameter AC form a semi-circle.
The angle in a semi-circle is always a right angle. It is the angle formed by the diameter on any point on the semi-circle.
Therefore, we get
$\angle ADC = 90^\circ$
The arc ABC and the diameter AC form a semi-circle.
Therefore, we get
$\angle ABC = 90^\circ$
The arc BCD and the diameter BD form a semi-circle.
Therefore, we get
$\angle BCD = 90^\circ$
The arc BAD and the diameter BD form a semi-circle.
Therefore, we get
$\angle BAD = 90^\circ$
Therefore, in the cyclic quadrilateral ABCD, we have
$\angle ADC = \angle ABC = \angle BDC = \angle BAD = 90^\circ$
Since all the interior angles of the cyclic quadrilateral ABCD are right angles, the cyclic quadrilateral ABCD is a rectangle.
Thus, the correct option is option (b).
Note: The quadrilateral given in the question is a cyclic quadrilateral. A cyclic quadrilateral is that quadrilateral whose four vertices lie on a circle. The opposite interior angles of a cyclic quadrilateral are supplementary angles. In other words, the sum of the opposite interior angles of a cyclic quadrilateral is 180 degrees.