Circles Theorem 9.5 | Class 9 2026 - 27 Prep | Proof and Logic | NCERT | Morning Star Academy

Описание: Class 9 Maths Chapter 9, Circles Class 9

Class 9 Maths 2026-27 | Theorem 9.5 Circles | Start Your Prep Early!

In this video, we provide a detailed, step-by-step handwritten proof of Theorem 9.5 from the Class 9 Maths NCERT chapter on Circles. We explain the logic behind "Equal chords of a circle subtend equal angles at the centre"

What You Will Learn:
Complete handwritten proof of Theorem 9.5.
How to prove Class 9 Circles Theorem 9.5
Logical explanation of Congruent Triangles within Circles.
Tips for presenting geometry proofs in your school exams.

Mathematical Breakdown:
Given: A circle with centre O and two equal chords AB and PQ (i.e., AB=PQ).
Construction:
Draw perpendiculars from the centre to the chords: OM⊥AB and ON⊥PQ.
Join the centre to one end of each chord: Join OA and OP (these are the radii).
To Prove: The chords are at equal distance from the centre, meaning OM=ON.
Methodology: We utilize the SSS (Side-Side-Side) Congruence Rule to prove that △AOM≅△BOM.
Mathematical Proof
Step A: Use the Perpendicular Bisector Property
We know that a perpendicular drawn from the centre of a circle to a chord bisects the chord.
Since OX⊥AB, then AX=21​AB.
Since OY⊥CD, then CY=21​CD.
Since we are given that AB=CD, their halves must also be equal:
1/2 ​AB = 1/2 PQ
⟹AM=PN…(Equation 1)
Step B: Prove Congruence of Triangles
Now, consider the two right-angled triangles formed: △AMO and △PNO.
Right Angle: ∠AMO=∠PNO (both 90∘ by construction).
Hypotenuse: AO = PO (radii of the same circle).
Side: AM = PN (proved in Equation 1).
By the RHS (Right angle-Hypotenuse-Side) congruence criterion:
△AMO≅△PNO
Step C: Conclusion (CPCT)
Since the triangles are congruent, their corresponding parts are equal:
OM=ON (by CPCT)

Theorem 9.4:    • Circles Theorem 9.4 | Class 9 2026 - 27  P...  
Theorem 9.3:    • Circles Theorem 9.3 | Class 9 | Proof and ...  
Theorem 9.2:    • Circles Theorem 9.2 | Class 9 | Proof and ...  
Theorem 9.1:    • Circles Theorem 9.1 | Class 9 | Proof and ...  
Exercise 9.1:    • Circles Ex 9.1 | Class 9 | NCERT Solutions...  

Textbooks Covered: While we follow the NCERT curriculum, these concepts are essential for students using:
R.D. Sharma
R.S. Aggarwal
S. Chand & Pearson Foundation

About Morning Star Academy: We are building a comprehensive digital library of mathematics solutions. Our mission is to provide high-quality, step-by-step guidance for students aiming for academic excellence and competitive success.

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Link to the NCERT Textbook: https://amzn.to/3OtPLGj
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0:00 Start
0:08 Recap
00:43 Th. 9.5, What does it say?
02:24 Construction - Why use dotted lines?
05:04 Given Identified, Construction mentioned
05:37 To prove - Th. 9.5
06:03 Proof of Th. 9.5
11:26 Th. 9.5 revision for exam

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