Question

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

Answer 1

$\huge{\text{\underline{Question:-}}}$

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

$\huge{\text{\underline{Solution:-}}}$

★ Point to remember:-

We have circle with centre O. There are two equal chords AB and CD intersecting at point E.

★Construction:-

• Draw OM ⊥ AB
• And ON ⊥ CD
• Join OE.

★ To prove that:-

• ∠OEM = ∠OEN

★ In △OME and △ONE

We have:-

∠OME = ∠ONE $\implies$( 90°)

OE = OE $\implies$(Common)

OM=ON$\implies$ (Equidistant from center)

★ Therefore, by RHS congruence rule:-

We have:-

$\implies$△OME ≅△ONE

$\implies$∠OEM = ∠OEN $\implies$(CPCT)

$\implies$Hence proved !

★ Note:- Kindly refer to the attachment for the diagram.

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$\huge{\text{\underline{Additional\: Information:-}}}$

★ Circle:- A circle is the locus of all points equidistant from a central point.

★ Radius:- The radius of a circle is the distance from the center of the circle to any point on its circumference.

★ Secant:- A line intersecting a circle at any two points is called secant.

★ Diameter:- A chord passing through the point of the circle, is called diameter.

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