Tuesday, November 12, 2019

Chord Length - Angle at Center Relationship

Angle Subtended by a Chord at the Center
Take a chord RS of a circle with center O. Join the endpoints of the chord to the center of the circle. The ∠ ROS is called the angle subtended by the chord RS at the center. Let us examine the relationship between the length of the chord and the angle subtended by it at the center. Longer is the chord, the bigger will be the angle subtended by it at the center. What happens when we have two or more chords of equal lengths? Let us see :

We are given two-chords AB and CD of equal length. These chords belong to the same circle with center O.

In triangles AOB and COD
            OA = OC (Radii of the circle)
            OB = OD (Radii of the circle)
            AB = CD (Given)
       So △ AOB = △ COD (by SSS Rule)
Hence,  ∠ AOB = ∠ COD
We can say that equal chords of a circle subtend equal angles at the center.

The converse of the above statement is also true i.e. if the angles subtended by chords of a circle at the center are equal, then the chords are equal.

What about the arc lengths corresponding to the chords: With the help of tracing paper, we can see that arc length AB = arc length CD.

जीवा द्वारा केन्द्र पर अंतरित कोण
O केन्द्र बिन्दु के किसी वृत्त की एक जीवा RS लीजिए और जीवा के बिन्दुओं R व S को केन्द्र बिन्दु से जोड़िए। इस प्रकार बना कोण ∠ ROS , जीवा RS द्वारा केन्द्र पर अंतरित कोण कहलाता है। आइए हम जीवा की लंबाई की माप और उसके द्वारा केन्द्र पर अंतरित कोण के बीच के संबंध की जाँच करें। जीवा की लंबाई अधिक होगी ,तो उसके द्वारा केन्द्र पर अंतरित कोण भी बड़ा होगा। क्या होगा यदि हम दो बराबर लंबाई की जीवाएं लें ? आइए देखते हैं :

त्रिभुज AOB और COD में
            OA = OC (एक ही वृत्त की त्रिज्याएं )
            OB = OD (एक ही वृत्त की त्रिज्याएं)
            AB = CD (दिया है)
  अत:   △ AOB = △ COD (SSS नियम से)
  इस प्रकार,∠ AOB = ∠ COD
हम कह सकते हैं कि वृत्त की बराबर जीवाएं केन्द्र पर बराबर कोण अंतरित करती हैं

उपर दिए कथन का विलोम भी सही है , यदि एक वृत्त की जीवाओं द्वारा केन्द्र पर अंतरित कोण बराबर हों , तो वे जीवाएं बराबर होती हैं।

यहां जीवाओं से जुड़े चाप की लंबाई के बारे में क्या कहा जा सकता है : एक ट्रेसिंग पेपर की मदद से हम यह देख सकते हैं कि AB चाप की लंबाई = CD चाप की लंबाई

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    1. It was a pleasure to learn this lesson from you, because your all topics are well explained and quite understandable.

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  3. You explained every point so well. I am also a mathematics teacher sometimes it's become so difficult to explain to students Chord Length- Angle at Center Relationship because each student has different abilities to understand and learn we have to explain each student differently. I'll try your way of explaining and pointing while giving them an explanation of Chord Length.

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