# Ex.10.6 Q7 Circles Solution - NCERT Maths Class 9

## Question

\(\begin {align} {AC}\end {align}\) and \(\begin {align} {BD}\end {align}\) are chords of a circle which bisect each other. Prove that

(i) \(\begin {align} {AC}\end {align}\) and \(\begin {align} {BD}\end {align}\) are diameters,

(ii) \(\begin {align} {ABCD}\end {align}\) is a rectangle.

## Text Solution

**What is known?**

\(\begin {align} {AC}\end {align}\) and \(\begin {align}{BD}\end {align}\) are **\(2\)** chords of a circle which bisect each other.

**What is unknown?**

Proof that \(\begin {align} {AC}\end {align}\) and \(\begin {align} {BD}\end {align}\) are diameters and \(\begin {align} {ABCD}\end {align}\) is a rectangle.

**Reasoning:**

- A quadrilateral \(\begin {align} {ABCD}\end {align}\) is called cyclic if all the four vertices of it lie on a circle.
- The sum of either pair of opposite angles of a cyclic quadrilateral is \(180^{\circ}.\)
- Opposite angles in a parallelogram are equal.
- Using Side-Angle-Side (
**SAS**criteria) and Corresponding parts of congruent triangles (**CPCT**) we prove the statement.

**Steps:**

Let \(\begin {align} {AC}\end {align}\) and \(\begin {align} {BD}\end {align}\) be **\(2\)** chords intersecting at \(\begin {align} {O.}\end {align}\)

In \(\begin {align} \Delta {AOB}\end {align}\) and \(\begin {align} \Delta {COD,}\end {align}\)

\(\begin{align}{OA}&={OC} \qquad \qquad (\text { Given }) \\ {OB}&={OD} \qquad\quad \;\;\, \text { (Given) } \\ \angle {AOB}&=\angle {COD} \qquad \; \text { (Vertically opposite angles) }\end{align}\)

\(\begin{align}∴ \Delta {AOB} &\cong \Delta {COD} \text { (SAS congruence rule) } \\ \qquad {AB}&={CD}({By} {CPCT})\end{align}\)

Similarly, it can be proved that \(\begin{align} \Delta {AOD} \cong \Delta {COB} \end{align}\)

\(\begin{align} ∴ {AD}={CB} \qquad (\text{By CPCT}) \end{align}\)

Since in quadrilateral \(\begin{align} {ABCD,} \end{align}\) opposite sides are equal in length, \(\begin{align} {ABCD} \end{align}\) is a parallelogram.

We know that opposite angles of a parallelogram are equal.

\(\begin{align} ∴ \angle {A}=\angle {C} \end{align}\)

\[\begin{align}\text { However, } \angle {A}+\angle {C}&=180^{\circ}({ABCD} \text { is a cyclic quadrilateral) } \\\angle {A}+\angle {A} &=180^{\circ} \\ 2 \angle {A}&=180^{\circ} \\ ∴ \angle {A}&=90^{\circ}\end{align}\]

As \(\begin{align}{ABCD}\end{align}\) is a parallelogram and one of its interior angles is \(90^{\circ,}\) therefore, it is a rectangle.

\(\begin{align}\angle {A}\end{align}\) is the angle subtended by chord \(\begin{align}\angle {BD.}\end{align}\)

And as .\(\begin{align}\angle {A} = 90^{\circ}, \end{align}\) therefore, \(\begin{align} {BD}\end{align}\) should be the diameter of the circle. Similarly, \(\begin{align} {AC}\end{align}\) is the diameter of the circle.