# **Critical Evaluation of Chapter 10: Circles (CBSE Class 9 Mathematics)**
## **1. Key Concepts in Circles**
1. **Definition of a Circle:** A set of all points in a plane that are equidistant from a fixed point called the center.
2. **Basic Terms in Circles:**
– **Radius (r):** Distance from the center to any point on the circle.
– **Diameter (d):** Twice the radius. \(d = 2r\).
– **Chord:** A line segment joining two points on a circle.
– **Secant:** A line that intersects a circle at two points.
– **Tangent:** A line that touches the circle at exactly one point.
– **Arc:** A portion of the circle’s circumference.
– **Sector:** Region enclosed by two radii and an arc.
– **Segment:** Region enclosed by a chord and an arc.
3. **Important Theorems:**
– **Theorem 1:** Equal chords of a circle subtend equal angles at the center.
– **Theorem 2:** The perpendicular from the center to a chord bisects the chord.
– **Theorem 3:** Equal chords are equidistant from the center.
– **Theorem 4:** The angle subtended by an arc at the center is twice the angle subtended at any point on the remaining part of the circle.
– **Theorem 5:** The sum of opposite angles of a cyclic quadrilateral is 180°.
– **Theorem 6:** The length of the tangent from an external point to a circle is equal.
—
## **2. Critical Evaluation of the Chapter**
### **(a) Conceptual Challenges Faced by Students**
1. **Difficulty Understanding Theorems and Proofs**
– Many students struggle to visualize circle theorems and their proofs.
– Example: Proving that tangents from an external point are equal requires a clear understanding of perpendicular bisectors.
2. **Confusion in Applying Theorems**
– Students often misapply angle properties, especially in cyclic quadrilaterals.
– Example: Misinterpreting that the angle at the center is **twice** the angle on the circumference.
3. **Understanding Tangents and Chords**
– Tangents and their properties, such as equal lengths, are often misused in solving numerical problems.
4. **Visualizing Geometrical Relationships**
– Many students struggle with diagrammatic interpretations, especially when multiple circles or chords are involved.
—
## **3. Evaluation of Past 10 Years’ CBSE Questions**
### **1. Definition-Based Questions**
#### **Question (2023, 2019, 2017, 2015)**
– Define a circle and explain the terms **radius, chord, secant, tangent, and arc**.
#### **Solution:**
– A **circle** is the set of all points equidistant from a fixed point (center).
– **Radius:** A segment from the center to the circumference.
– **Chord:** A segment joining two points on the circle.
– **Secant:** A line cutting the circle at two points.
– **Tangent:** A line touching the circle at exactly one point.
– **Arc:** A curved portion of the circumference.
—
### **2. Theorem-Based Questions**
#### **Question (2022, 2018, 2016, 2013)**
– State and prove: *The perpendicular from the center to a chord bisects the chord.*
#### **Solution:**
– Given: A circle with center \(O\), and a chord \(AB\).
– To Prove: \(OD \perp AB\) bisects \(AB\) at \(D\).
– Proof:
– Join \(OA\) and \(OB\).
– In \(\triangle OAD\) and \(\triangle OBD\):
– \(OA = OB\) (radii of the same circle).
– \(OD = OD\) (common side).
– \(\angle ODA = \angle ODB = 90^\circ\) (given).
– By **RHS Congruence Theorem**, \(\triangle OAD \cong \triangle OBD\).
– \(AD = BD\).
✅ **Hence proved**.
—
### **3. Numerical Questions on Chords & Tangents**
#### **Question (2021, 2019, 2014)**
– The radius of a circle is **10 cm**. A chord is **16 cm** long. Find the perpendicular distance of the chord from the center.
#### **Solution:**
Using the theorem: *The perpendicular from the center bisects the chord.*
– Given: \(OA = 10 cm\), \(AB = 16 cm\), \(AD = 8 cm\)
– Using Pythagoras theorem in \(\triangle OAD\):
\[
OD^2 + AD^2 = OA^2
\]
\[
OD^2 + 8^2 = 10^2
\]
\[
OD^2 + 64 = 100
\]
\[
OD^2 = 36
\]
\[
OD = 6 cm
\]
✅ **Answer: 6 cm**
—
### **4. Angle Property Questions**
#### **Question (2020, 2018, 2016)**
– In a circle, an arc subtends an angle **70°** at the center. Find the angle subtended at the circumference.
#### **Solution:**
Using the theorem: *The angle at the center is twice the angle at the circumference.*
\[
\theta = \frac{1}{2} \times 70^\circ = 35^\circ
\]
✅ **Answer: 35°**
—
### **5. Application-Based Questions**
#### **Question (2019, 2015, 2012)**
– Two tangents are drawn from an external point **P** to a circle with center **O**. Prove that these tangents are equal in length.
#### **Solution:**
– Given: \(PA\) and \(PB\) are tangents from \(P\) to the circle.
– To Prove: \(PA = PB\).
– Proof:
– Join \(OA, OB, OP\).
– \(OA = OB\) (radii of the same circle).
– \(\angle OAP = \angle OBP = 90^\circ\) (tangent is perpendicular to radius).
– In \(\triangle OAP\) and \(\triangle OBP\):
– \(OA = OB\)
– \(OP = OP\) (common side)
– \(\angle OAP = \angle OBP = 90^\circ\)
– By **RHS Congruence Theorem**, \(\triangle OAP \cong \triangle OBP\).
– \(PA = PB\) (corresponding sides).
✅ **Hence proved**.
—
## **4. Exam Preparation Tips**
1. **Memorize All Theorems:** Understanding and proving theorems is critical.
2. **Practice Diagram-Based Questions:** Many CBSE questions are based on geometrical constructions.
3. **Work on Angle and Chord Problems:** Learn how to apply theorems logically in problem-solving.
4. **Solve Past Year Papers:** Focus on numerical and proof-based questions.
Would you like a diagram for the **tangent theorem** or any other concept from this chapter? PLZ COMENT……