Chapter 11: Constructions (CBSE Class 9 Mathematics)
## **1. Key Concepts in Constructions**
Chapter 11 of CBSE Class 9 Mathematics focuses on **geometrical constructions** using a ruler and compass. This chapter is crucial as it builds the foundation for higher-level geometry and practical applications.
### **Topics Covered:**
1. **Basic Constructions**
– Constructing a bisector of a given angle.
– Constructing the perpendicular bisector of a line segment.
2. **Construction of Triangles**
– Given base, base angle, and sum of the other two sides.
– Given base, base angle, and difference of the other two sides.
– Given perimeter and two base angles.
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## **2. Critical Evaluation of the Chapter**
### **(a) Common Challenges Faced by Students**
1. **Difficulty in Maintaining Precision:**
– Errors in measurements lead to incorrect constructions.
– Slight mistakes in using a compass result in inaccurate figures.
2. **Misinterpretation of Given Data:**
– Confusion between sum/difference of sides in triangle constructions.
– Misplacing the base angle or using the wrong approach.
3. **Errors in Using a Compass and Ruler Together:**
– Students often struggle with drawing perpendicular bisectors accurately.
4. **Forgetting Theoretical Justifications:**
– Many students focus on drawing but fail to justify their steps logically.
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## **3. Evaluation of Past 10 Years’ CBSE Questions**
### **1. Basic Construction-Based Questions**
#### **Question (2023, 2020, 2017, 2015)**
– Construct an angle of **75°** using a compass and bisector method.
#### **Solution:**
1. Draw a ray \( OA \).
2. Using a compass, mark an arc from \( O \) that cuts \( OA \) at \( B \).
3. Without changing the compass width, place it at \( B \) and draw another arc to get a point \( C \).
4. Keeping the compass at \( C \), mark another arc to get point \( D \).
5. Bisect \( \angle ABC \) to get \( 75^\circ \).
✅ **Final Answer: Angle of 75° Constructed**
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### **2. Triangle Construction Questions**
#### **Question (2022, 2018, 2014)**
– Construct a **triangle** where the base is **7 cm**, base angle is **50°**, and the sum of the other two sides is **12 cm**.
#### **Solution:**
1. Draw the base \( BC = 7 cm \).
2. Draw \( \angle B = 50^\circ \).
3. Extend the line beyond \( B \).
4. From \( B \), draw an arc of **12 cm** along the extended line.
5. Connect this new point with \( C \).
6. Bisect this line to locate the third vertex.
✅ **Final Answer: Triangle Constructed Successfully**
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### **3. Construction with Perpendicular Bisectors**
#### **Question (2021, 2019, 2016)**
– Construct a **perpendicular bisector** for a line segment of **8 cm**.
#### **Solution:**
1. Draw a line \( AB = 8 cm \).
2. Using a compass, place it at \( A \) and draw arcs above and below the line.
3. Without changing the radius, repeat the process from \( B \).
4. Mark the intersection points and draw a perpendicular bisector.
✅ **Final Answer: Perpendicular bisector drawn correctly**
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### **4. Application-Based Questions**
#### **Question (2020, 2015, 2012)**
– Given the perimeter of a **triangle = 15 cm** and base angles **45° and 60°**, construct the triangle.
#### **Solution:**
1. Draw a line segment equal to **15 cm (perimeter)**.
2. Construct **45°** and **60°** at each end.
3. The intersection of these two rays gives the required triangle.
✅ **Final Answer: Triangle Constructed Using Perimeter**
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## **4. Exam Preparation Tips**
1. **Practice with Precision:** Small errors can affect the accuracy of the construction.
2. **Revise All Triangle Construction Methods:** Understand when to apply each method.
3. **Use Proper Justifications:** Always explain why a step is performed.
4. **Solve Past Year Papers:** This helps in identifying common patterns.
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Would you like a step-by-step construction diagram for any of these problems? 😊
