NCERT Solution for Class8 Maths Chapter4: Practical Geometry
Practical Geometry is the fourth chapter of the Class8 Mathematics textbook prescribed by the NCERT. It introduces the concepts of points, lines, line segments, rays, and angles, and shows how to construct basic geometric figures using a straightedge and a compass. For students preparing for board examinations, having clear, stepbystep solutions is crucial. The following article provides a concise yet comprehensive guide to the NCERT solutions for this chapter, covering the theory, solving strategies, and workedout examples.
Why a Dedicated NCERT Solution Matters
The NCERT textbook is designed with a logical progression of ideas. However, the wording of some problems can be confusing for beginners. A wellstructured solution does three things:
- Clarifies concepts: Each solution restates the key idea before diving into calculations.
- Shows the method: It demonstrates the construction steps or algebraic manipulations expected in the exam.
- Provides shortcuts: Where applicable, it suggests quicker ways to obtain the required result, saving valuable time during a test.
Chapter Overview Topics Covered
The chapter is divided into two sections:
- Fundamental Concepts: Points, lines, line segments, rays, and angles (including types of angles and the angle sum property).
- Construction Problems: Drawing perpendiculars, bisectors, triangles, and other figures using a compass and straightedge.
Key Definitions
Point: A location in space with no dimension.
Line: An infinite set of points extending in both directions, denoted by a lowercase letter or two points on the line (e.g., AB).
Line Segment: A part of a line bounded by two endpoints (e.g., AB).
Ray: A part of a line that starts at an endpoint and extends infinitely in one direction (e.g., AB where A is the endpoint).
Angle: The region formed by two rays sharing a common endpoint (the vertex). Angles are measured in degrees.
Important Theorems
- Angle Sum Property: The sum of the interior angles of a triangle is 180.
- Complementary & Supplementary Angles: Two angles are complementary if their sum is 90, and supplementary if their sum is 180.
- Vertical Angles: When two lines intersect, the opposite (nonadjacent) angles are equal.
Solving the Exercise Questions StepbyStep Approach
1. Determining Unknown Angles
Many problems ask you to find the value of a missing angle when other angles in the figure are known. Follow this sequence:
- Identify the type of angle (right, acute, obtuse, etc.).
- Use the relevant theorem (e.g., angle sum of a triangle, linear pair, vertical angle).
- Set up an equation and solve for the unknown.
Example: In triangle ABC, A = 40 and B = 55. Find C.
Solution: A + B + C = 180 40 + 55 + C = 180 C = 180 95 = 85.
2. Construction Problems
Construction questions test the ability to use a compass and a ruler. While the NCERT textbook provides diagrams, the solution must explain each step clearly.
General Procedure for a Construction
- Read the problem statement carefully; note the given lengths and angles.
- Draw the base line or segment as specified.
- Use the compass to copy lengths or to mark arcs for constructing equal angles.
- Connect the resulting points with a straightedge to complete the figure.
- Verify the construction by checking all given measurements.
Example: Draw a triangle ABC such that AB = 5cm, AC = 7cm and BAC = 60.
Solution:
- Draw a base lineAB of length 5cm.
- Place the compass at A and set the radius to 7cm. Draw an arc above AB.
- With the compass still at A, draw a 60 angle using the protractor (or construct a 60 angle by constructing an equilateral triangle on the base AB).
- The point where the 7cm arc meets the 60 ray is point C. Connect B to C.
- Measure BAC to confirm it is 60 and verify AB = 5cm, AC = 7cm.
3. Perpendicular and Angle Bisector Constructions
Two of the most frequently asked constructions are the perpendicular from a point to a line and the bisector of an angle.
Perpendicular from a Point to a Line
- Place the compass at the given point P and draw an arc intersecting the line at two points, say Q and R.
- Without changing the compass width, draw arcs centered at Q and R that intersect above and below the line.
- Join the intersection points of these arcs; the resulting line through P is the required perpendicular.
Angle Bisector
- Place the compass at the vertex of the angle and draw an arc cutting both sides of the angle at points M and N.
- Keeping the same radius, draw arcs centered at M and N; they intersect at a point P inside the angle.
- Draw a straight line from the vertex through P; this line bisects the angle.
Common Mistakes and Tips
- Misreading the diagram: Always label all given points before starting calculations.
- Using wrong units: The NCERT textbook expects measurements in centimeters for construction problems.
- Skipping verification: After a construction, doublecheck all given lengths and angles; a small error in the early step can propagate.
- Forgetting that a straight line has infinite length: In prooftype questions, state that a line extends indefinitely unless a specific segment is mentioned.
Sample Questions with Full Solutions
Q.1 Find the value of x
In the figure, A = 2x, B = 3x 20, and they form a linear pair.
Because they are a linear pair, A + B = 180.
2x + (3x 20) = 180 5x 20 = 180 5x = 200 x = 40.
Thus, A = 80 and B = 100.
Q.2 Construction of a RightAngled Triangle
Construct a rightangled triangle with base = 6cm, height = 8cm, and the right angle at the origin.
- Draw a horizontal line AB of length 6cm. Mark point A as the origin.
- From A, draw a vertical line AC upwards. Use a compass set to 8cm to mark point C on this line.
- Join B to C. A is a right angle because AB is horizontal and AC is vertical.
- Measure AB = 6cm, AC = 8cm, and verify that BC = 10cm (by Pythagoras, 6 + 8 = 10).
Q.3 Proving Two Angles Are Equal
Given two intersecting lines forming vertical angles 1 and 2, prove that 1 = 2.
By definition, vertical angles are the opposite angles formed when two lines intersect. Hence, they are equal. This property is a direct consequence of the linear pair axiom.
How to Use These Solutions Effectively
While the solutions above provide a clear roadmap, students should practice the following strategies to retain the concepts:
- Active rewriting: Copy the solution in your own words while studying, which reinforces understanding.
- Timed practice: Solve a set of problems within a fixed period to improve speed for board exams.
- Peer teaching: Explain a construction to a classmate; teaching cements knowledge.
Additional Resources
For deeper practice, the following resources complement the NCERT solutions:
- Official NCERT PDF Download the original textbook for reference.
- Khan Academy Geometry Video tutorials on constructing angles and triangles.
- GeoGebra Interactive Applet Experiment with constructions digitally.
Conclusion
The Practical Geometry chapter builds a foundation for higherlevel geometry and trigonometry. By mastering the definitions, theorems, and construction techniques presented in the NCERT solution, students can approach both classroom exercises and boardexam questions with confidence. Consistent practice, careful verification, and the habit of writing clear, stepbystep solutions will ensure that the concepts become second nature, paving the way for success in mathematics.
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