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

9th Class Maths – NCERT Important Questions

Ans:

Step 1: Let’s make a diagram for better understanding of the question:

Here AB and PQ are the 2 equal chords, R is the point of intersection and line OR is making ∠ ORB with chord AB and ∠ ORP with chord PQ.

We need to prove that ∠ ORB = ∠ ORP

Step 2:

Let’s draw perpendicular OC on chord AB and perpendicular OD on chord PQ.

Now let’s compare Δ OCR and Δ ODR:

∠OCR = ∠ ODR          (∵ both angles are 90⁰)

OC = OD                     (∵ Equal chords are at equal distance from Center)

OR = OR                     (∵ OR is common side to both triangles)

∴ Δ OCR Δ ODR

Step 3: Now by Corresponding Parts of Congruent Triangles (CPCT) rule:

∠ ORC = ∠ ORD

∴ ∠ ORB = ∠ ORP

Hence Proved!

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