Every Theorem on Circle with Proofs.| Theorem on Circles.| Class 9 |NCERT.

Summary

This video explores key theorems related to circles, offering clear proofs for each. It covers topics such as equal chords subtending equal angles at the center, perpendiculars bisecting chords, equal chords being equidistant from the center, and more. The video also delves into relationships between angles, segments, and arcs, including the proof that the angle subtended by an arc at the center is double the angle subtended at any point on the remaining circle. It concludes by demonstrating that only one unique circle can pass through three non-collinear points, with the final focus on angles formed in the same segment of a circle.

Takeaways

• 😀 Equal chords in a circle subtend equal angles at the center.
• 😀 Theorem: A perpendicular drawn from the center of a circle to a chord bisects the chord.
• 😀 Equal chords in a circle are equidistant from the center.
• 😀 The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.
• 😀 The angle in a semicircle is always 90°.
• 😀 Only one unique circle can pass through three non-collinear points.
• 😀 Angles formed in the same segment of a circle are equal.
• 😀 The proof of equal chords subtending equal angles relies on congruent triangles using the SSS rule.
• 😀 The congruency of triangles is key in proving that a perpendicular from the center bisects a chord.
• 😀 The concept of central angles and their relationship to the angles subtended at other points on the circle is critical in understanding circle theorems.

Q & A

What does the theorem about equal chords subtending equal angles at the center state?

The theorem states that if two chords of a circle are equal in length, they subtend equal angles at the center of the circle. This is proven by showing that the triangles formed by joining the center to the endpoints of the chords are congruent.

What is the converse of the theorem about equal chords subtending equal angles?

The converse states that if two chords subtend equal angles at the center, then the chords must be equal in length. This is proven by showing that the triangles formed by the center and the chord endpoints are congruent, leading to equal lengths for the chords.

How is it proven that a perpendicular from the center to a chord bisects the chord?

It is proven using congruent triangles. The line from the center is perpendicular to the chord, creating two right triangles. By using the RHS (Right-Angle-Hypotenuse-Side) congruency rule, it is shown that the perpendicular bisects the chord.

What does the theorem about equal chords being equidistant from the center say?

This theorem states that if two chords of a circle are equal in length, they are equidistant from the center of the circle. The proof involves constructing perpendiculars from the center to each chord and showing that these perpendiculars are equal.

How does the theorem about the angle at the center being double the angle at any point on the circle work?

This theorem states that the angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference of the circle. The proof involves using isosceles triangles and properties of exterior angles.

Why is the angle in a semicircle always 90°?

The angle subtended by a diameter of a circle at any point on the semicircle is always 90°. This is a result of the previous theorem, where the angle at the center is double the angle at the circumference, and for a semicircle, this leads to a right angle.

How can three non-collinear points uniquely determine a circle?

Any three non-collinear points lie on a unique circle. This is proven by constructing the perpendicular bisectors of the chords formed by these points. The intersection of these bisectors gives the center of the circle, and the radius is the distance from the center to any of the points.

What does the theorem about angles in the same segment of a circle state?

This theorem states that angles subtended by the same chord in the same segment of a circle are equal. This is proved by showing that the angle at the center of the circle is the same for any point on the segment, and thus the angles at the circumference must be equal.

What is the significance of constructing perpendicular bisectors in circle geometry?

Perpendicular bisectors are key in many circle-related theorems, such as proving that equal chords are equidistant from the center or that three non-collinear points uniquely define a circle. They help in finding the center and establishing congruency between triangles.

How does the SAS rule help in proving congruency in circle geometry?

The SAS (Side-Angle-Side) rule is used to prove the congruency of triangles by showing that two sides and the included angle are equal. In circle geometry, it helps in proving the equality of chords and the bisecting of chords by the perpendicular from the center.