# Ex.10.5 Q7 Circles Solution - NCERT Maths Class 9

## Question

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

## Text Solution

**What is known?**

Diagonals of a cyclic quadrilateral are diameters of a circle passing through the vertices.

**What is unknown?**

To prove lines joining the vertices is a rectangle.

**Reasoning:**

- The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
- The sum of either pair of opposite angles of a cyclic quadrilateral is \(180^{\circ}.\)
- Diameter is a chord.

**Steps:**

Let \({DB}\) be the diameter of the circle which is also a chord.

Then \(\begin {align} \angle {BOD}=180^{\circ} \end {align}\)

We know that, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

\[\begin{align} \angle {BAD} &=\frac{1}{2} \times \angle {BOD} \\ &=90^{\circ} \end{align}\]

Similarly, \(\begin {align} \angle {BCD}=90^{\circ} \end {align}\)

Now considering \(\text{AC}\) as the diameter of the circle, we get \(\begin {align} \angle {AOC}=180^{\circ} \end {align}\)

We know that, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

\[\begin{align} \angle {ABC} &=\frac{1}{2} \times \angle {AOC} \\ &=90^{\circ} \end{align}\]

Similarly, \(\begin {align} \angle {ADC}=90^{\circ} \end {align}\)

As you can see, all the angles at the corners are \(90^\circ \)we can say that the shape joining the vertices is a rectangle.

This problem can also be solved by using the property of cyclic quadrilaterals.