# Ex.10.4 Q3 Circles Solution - NCERT Maths Class 9

## Question

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the center makes equal angles with the chords.

## Text Solution

**What is known?**

Chords are equal. Chords intersect at a point within the circle.

**What is unknown?**

Line joining the point of intersection to the centre makes equal angles with the chords.

**Reasoning:**

Equal chords are equidistant from the centre. Using this and Right angled- Hypotenuse-Side (RHS) criteria and Corresponding parts of congruent triangles (CPCT) we prove the statement.

**Steps:**

Let \({AB}\) and \({CD}\) be the **\(2\)** equal chords. \({AB = CD.}\)

Let the chords intersect at point \({E}\). Join \({OE.}\)

Draw perpendiculars from the centre to the chords. Perpendicular bisects the chord \(AB\) at \(M\) and \(CD\) at \(N\).

**To prove:** \(\angle {OEN}=\angle {OEM}\)

\({\rm{In}} \;\Delta {OME}\; {\rm{and}} \;\Delta {ONE}\) \[\begin{align} \angle {M}&=\angle {N}=90^{\circ}\\ {OE}&={OE}\\ {OM}&={ON} \quad \\\end{align}\]

( Equal chords are equidistant from the centre.)

By **RHS** criteria, \(\Delta {OME}\) and \(\Delta {ONE}\) are congruent.

\(\text{So by CPCT, }\angle {OEN}=\angle {OEM}\)

Hence proved that line joining the point of intersection of **\(2\)** equal chords to the centre makes equal angles with the chords.