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Ex.10.4 Q3 Circles Solution - NCERT Maths Class 9

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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.

 Video Solution
Circles
Ex 10.4 | Question 3

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.

 Video Solution
Circles
Ex 10.4 | Question 3
  
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