जीवा द्वारा एक बिंदु पर अंतरित कोण: Difference between revisions

Revision as of 14:59, 23 August 2024

Theorem 1 : Equal chords of a circle subtend equal angles at the centre.

Proof :Consider a circle and draw two equal chords $AB$ and $CD$ of a circle with center $O$ as shown in the figure 1.

Fig.1

We want to prove that : $\angle AOB=\angle COD$

From the triangles, $AOB$ and $COD$ , we get

$OA=OC$ (Radii of a circle)

$OB=OD$ (Radii of a circle)

$AB=CD$ (Given)

By, using Side-Side-Side (SSS Rule), we can write:

$\triangle AOB\cong \triangle COD$

As the triangles are congruent, the angles should be of equal measurement.

Therefore, $\angle AOB=\angle COD$ [Using Corresponding parts of the congruent triangle (CPCT)]

Hence, the above theorem is proved.

Theorem 2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

@@ Line 1: / Line 1: @@
+'''Theorem 1''' : Equal chords of a circle subtend equal angles at the centre.
+'''Proof''' :Consider a circle and draw two equal chords <math>AB</math> and <math>CD</math> of a circle with center <math>O</math> as shown in the figure 1.
+[[File:Angle-subtend-chord-point.jpg|alt=Angle-subtend-chord-point|thumb|Fig.1]]
+We want to prove that : <math>\angle AOB = \angle COD </math>
+From the triangles, <math>AOB </math> and <math>COD </math>, we get
+<math>OA=OC</math> (Radii of a circle)
+<math>OB=OD</math> (Radii of a circle)
+<math>AB=CD</math> (Given)
+By, using Side-Side-Side (SSS Rule), we can write:
+<math>\triangle AOB \cong \triangle COD </math>
+As the triangles are congruent, the angles should be of equal measurement.
+Therefore, <math>\angle AOB = \angle COD </math> [Using Corresponding parts of the congruent triangle (CPCT)]
+Hence, the above theorem is proved.
+'''Theorem 2''' : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
 [[Category:वृत्त]][[Category:कक्षा-9]][[Category:गणित]]
-Angle subtended by a chord at a point