Circles Class 9 All Theorems

Circles: A Comprehensive Guide to Class 9 Theorems

Understanding circles is a crucial step in your geometrical journey. This comprehensive guide delves into all the essential theorems related to circles typically covered in Class 9 mathematics. We'll explore each theorem, provide detailed explanations, and illustrate their applications with examples. This guide aims to not only help you understand the theorems but also to build a strong foundation for more advanced geometrical concepts. By the end, you'll be confident in tackling any circle-related problem.

Introduction to Circles and Basic Definitions

Before diving into the theorems, let's refresh some fundamental definitions related to circles:

• Circle: A circle is a set of all points in a plane that are equidistant from a fixed point called the center.
• Radius: The distance from the center of the circle to any point on the circle.
• Diameter: A line segment passing through the center and connecting two points on the circle. It is twice the length of the radius.
• Chord: A line segment connecting any two points on the circle.
• Secant: A line that intersects the circle at two points.
• Tangent: A line that touches the circle at exactly one point, called the point of tangency.
• Arc: A portion of the circumference of the circle.
• Segment: A region bounded by a chord and an arc.
• Sector: A region bounded by two radii and an arc.

Theorem 1: The Perpendicular from the Centre to a Chord Bisects the Chord

This theorem states that a perpendicular drawn from the center of a circle to a chord bisects the chord. In simpler terms, it cuts the chord into two equal parts.

Explanation:

Consider a circle with center O and a chord AB. Let OM be the perpendicular drawn from O to AB. Then, according to this theorem, AM = MB.

Proof:

We can prove this using congruent triangles. Consider triangles OMA and OMB. We have:

• OM = OM (common side)
• ∠OMA = ∠OMB = 90° (given that OM is perpendicular to AB)
• OA = OB (radii of the same circle)

By RHS (Right-hand side) congruence criterion, ΔOMA ≅ ΔOMB. Therefore, AM = MB.

Application: This theorem is frequently used to find the length of a chord or the distance of the chord from the center.

Theorem 2: The Line Joining the Centre to the Midpoint of a Chord is Perpendicular to the Chord

This theorem is the converse of the previous one. It states that if a line drawn from the center of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

Explanation:

Again, consider a circle with center O and chord AB. If M is the midpoint of AB, then the line OM is perpendicular to AB (∠OMA = ∠OMB = 90°).

Proof:

This can also be proven using congruent triangles. Since M is the midpoint of AB, AM = MB. Also, OA = OB (radii). And OM is a common side. By SSS (Side-Side-Side) congruence, ΔOMA ≅ ΔOMB. Therefore, ∠OMA = ∠OMB. Since these angles are supplementary and equal, they must each be 90°. Hence, OM is perpendicular to AB.

Application: This theorem helps in determining if a line segment connecting the center to a point on a chord is perpendicular to that chord.

Theorem 3: Equal Chords of a Circle are Equidistant from the Centre

This theorem establishes a relationship between the lengths of chords and their distances from the center.

Explanation:

If two chords, AB and CD, in a circle are equal in length (AB = CD), then their distances from the center are also equal. Let OM and ON be the perpendicular distances from the center O to chords AB and CD respectively. Then, OM = ON.

Proof:

Draw perpendiculars OM and ON from the center O to chords AB and CD respectively. According to Theorem 1, AM = MB = AB/2 and CN = ND = CD/2. Since AB = CD, we have AM = CN. Now consider right-angled triangles OMA and ONC. We have:

• OA = OC (radii)
• AM = CN (proven above)

By Pythagoras theorem, OM² + AM² = OA² and ON² + CN² = OC². Since OA = OC and AM = CN, we get OM² = ON². Therefore, OM = ON.

Application: This theorem is useful for comparing the lengths of chords based on their distances from the center.

Theorem 4: Chords of a Circle Which are Equidistant from the Centre are Equal

This is the converse of Theorem 3.

Explanation:

If two chords, AB and CD, are equidistant from the center (OM = ON, where OM and ON are perpendicular distances from the center to the chords), then the chords are equal in length (AB = CD).

Proof:

The proof is similar to the converse of Theorem 3. Using the Pythagorean theorem on right-angled triangles OMA and ONC, and knowing OM = ON, we can conclude that AM = CN. Since AM = AB/2 and CN = CD/2, we conclude AB = CD.

Application: This theorem aids in determining the equality of chords based on their distances from the circle's center.

Theorem 5: Angle Subtended by an Arc at the Centre is Double the Angle Subtended by the Same Arc at Any Point on the Remaining Part of the Circle

This theorem relates angles subtended at the center and at any point on the circumference.

Explanation:

Consider an arc AB subtending an angle ∠AOB at the center O and an angle ∠APB at a point P on the remaining part of the circle. Then, ∠AOB = 2∠APB.

Proof:

There are several cases to consider depending on the position of P relative to the arc AB. The proof typically involves considering different triangles and using angle properties. A common approach is to consider three cases: when P lies on the major arc, on the minor arc, and when P lies on the diameter. Each case involves constructing appropriate lines and using the properties of isosceles triangles and exterior angles. This proof is more intricate and requires a deeper understanding of angle properties.

Application: This theorem is fundamental in solving problems related to angles in circles. It's particularly helpful when dealing with cyclic quadrilaterals (discussed below).

Theorem 6: Angles in the Same Segment of a Circle are Equal

This theorem focuses on angles subtended by the same arc in the same segment.

Explanation:

If points A and B are fixed on a circle, then the angle subtended by the arc AB at any point on the remaining part of the circle is the same. In other words, ∠APB = ∠AQB for any points P and Q on the same arc.

Proof:

This is a direct consequence of Theorem 5. Since both angles ∠APB and ∠AQB are subtended by the same arc AB, and both are half of the angle subtended by the arc at the center, they must be equal.

Application: This theorem simplifies calculations involving angles in a circle, especially those related to cyclic quadrilaterals.

Theorem 7: The Opposite Angles of a Cyclic Quadrilateral are Supplementary

A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle.

Explanation:

In a cyclic quadrilateral ABCD, the sum of opposite angles is 180°. Therefore, ∠A + ∠C = 180° and ∠B + ∠D = 180°.

Proof:

This proof uses Theorem 5. Consider the arc ABC subtending ∠ADC at point D and the arc BCD subtending ∠BAD at point A. The angles subtended at the center by these arcs are double the angles at points D and A. The sum of the angles at the center formed by these two arcs is 360°. Therefore, their half, i.e., ∠ADC + ∠BAD, will be 180°. Similarly, we can prove for the other pair of opposite angles.

Application: This theorem is a powerful tool for solving problems involving cyclic quadrilaterals. It allows us to find unknown angles based on the known angles.

Theorem 8: The Alternate Segment Theorem

This theorem relates the angle between a chord and a tangent to the angle in the alternate segment.

Explanation:

Let a tangent be drawn at point A on a circle, and let a chord AB be drawn from point A. Let the angle between the tangent and the chord be θ. Then, the angle subtended by the chord AB in the alternate segment is also θ.

Proof:

The proof of this theorem uses Theorem 5 and the angle properties of triangles. By constructing a line through the center and another point on the circle, we can demonstrate the relationship between the angles. The details of the proof are beyond the scope of a simple explanation.

Application: This theorem is crucial for finding unknown angles when a tangent and a chord are involved.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a chord and a diameter?

A chord connects any two points on the circle, while a diameter is a special chord that passes through the center of the circle. A diameter is the longest possible chord.

Q2: Can a tangent intersect a circle at more than one point?

No, by definition, a tangent touches the circle at exactly one point.

Q3: What is a cyclic quadrilateral?

A cyclic quadrilateral is a quadrilateral whose four vertices lie on a single circle.

Q4: How do I know if a quadrilateral is cyclic?

A quadrilateral is cyclic if its opposite angles are supplementary (add up to 180°).

Conclusion

Mastering circle theorems is a significant milestone in your mathematical journey. These theorems provide the building blocks for understanding more complex geometrical concepts. Remember to practice regularly and apply these theorems to various problems. The more you work with these theorems, the more intuitive they will become. Don't hesitate to revisit these explanations and proofs as needed. With consistent effort, you will confidently navigate the world of circles and their related properties.