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Class 10th Mathematics Chapter 7 A TALE OF THREE INTERSECTING LINES

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I. Chapter Summary:

This chapter explores the properties and relationships of three important lines in a triangle: the altitude, median, and angle bisector. Students learn how these lines intersect within the triangle, their definitions, and how they help in understanding the geometry of triangles. The chapter also introduces special points like the orthocenter, centroid, and incenter, which are points of concurrency formed by these lines.

II. Key Concepts Covered:

• Altitude of a Triangle: A perpendicular line segment from a vertex to the opposite side (or its extension).

• Median of a Triangle: A line segment joining a vertex to the midpoint of the opposite side.

• Angle Bisector: A line dividing the angle at a vertex into two equal angles.

• Point of Concurrency: A point where three or more lines intersect.

• Orthocenter: The point where all three altitudes meet.

• Centroid: The point where all three medians intersect; also the center of mass of the triangle.

• Incenter: The point where all three angle bisectors intersect; center of the inscribed circle.

• Properties of these lines and points: Their significance in triangle geometry and real-world applications.

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

1. The three medians of a triangle meet at a point called:
   a) Orthocenter
   b) Centroid
   c) Incenter
   d) Circumcenter
   Answer: b) Centroid
   (PYQ 2020)

2. The altitude of a triangle is:
   a) A line bisecting the opposite side
   b) A line perpendicular to a side from the opposite vertex
   c) A line bisecting an angle
   d) None of the above
   Answer: b) A line perpendicular to a side from the opposite vertex
   (PYQ 2019)

3. The point of concurrency of angle bisectors is called:
   a) Centroid
   b) Orthocenter
   c) Incenter
   d) Circumcenter
   Answer: c) Incenter
   (PYQ 2018)

4. The medians of a triangle divide it into:
   a) Two equal parts
   b) Three equal parts
   c) Six equal parts
   d) Four equal parts
   Answer: c) Six equal parts
   (PYQ 2021)

(B) Short Answer Questions (2/3 Marks):

1. Define altitude of a triangle with a diagram.

2. What is a median? How is it constructed?

3. Explain the significance of the centroid in a triangle.

4. Describe the angle bisector and its point of concurrency.

(C) Long Answer Questions (5 Marks):

1. Prove that the three medians of a triangle intersect at a single point (centroid).

2. Describe the properties of orthocenter and centroid.

3. Explain how to construct the incenter of a triangle.

4. Write the differences between altitude, median, and angle bisector.

(D) HOTS (Higher Order Thinking Skills) Questions:

1. Can the orthocenter lie outside the triangle? Justify with an example.

2. Show how the centroid divides each median into two segments in the ratio 2:1.

IV. Key Formulas/Concepts:

• Centroid divides median in the ratio 2:1 (from vertex to midpoint).

• Altitude is perpendicular to the opposite side.

• Angle bisector theorem: The angle bisector divides the opposite side in the ratio of the adjacent sides.

V. Deleted Portions (CBSE 2025–2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026):

VII. Previous Year Questions (PYQs):

• 1 Mark: Definitions and identification (2018, 2019)

• 2/3 Marks: Construction and properties of medians and altitudes (2020, 2021)

• 5 Marks: Proofs and detailed explanations on concurrency points (2019, 2022)

VIII. Real-World Application Examples to Connect with Topics:

• Engineering designs involving triangular structures use centroid for stability.

• Construction and architecture rely on medians and centroids for balance points.

• Navigation and triangulation methods use altitude and angle bisectors for locating positions.

• Robotics and mechanical systems use geometric concurrency points for motion and balance.

IX. Student Tips & Strategies for Success (Class-Specific):

• Practice accurate constructions with a ruler and compass for medians, altitudes, and bisectors.

• Visualize concurrency points with multiple examples for better understanding.

• Solve a variety of problems involving concurrency and line properties in triangles.

• Use color-coded diagrams to differentiate medians, altitudes, and angle bisectors while studying.

X. Career Guidance & Exploration (Class-Specific):

• For Classes 9–10:

  • Builds foundational geometry knowledge vital for engineering, architecture, and design fields.

  • Prepares students for Olympiads and NTSE focusing on geometry.

• For Classes 11–12:

  • Advanced studies in geometry, trigonometry, and CAD software for engineering and design.

  • Important for entrance exams in engineering (JEE), architecture (NATA), and design.

XI. Important Notes:

• Refer regularly to the official CBSE and NCERT guidelines for syllabus updates.

• Focus on both theoretical understanding and practical constructions.

• Clear concept clarity helps in solving complex geometry problems efficiently.