CBSE Class VIII Mathematics NCERT Solutions
Practical Geometry (Exercise4.4)
This page provides a clear, stepbystep guide to solving the practical geometry questions that appear in Exercise4.4 of the NCERT ClassVIII Mathematics textbook. The aim is to help students understand the concepts, develop a systematic solving approach, and practice with workedout examples.
What is Practical Geometry?
Practical geometry deals with the application of basic geometric ideaspoints, lines, line segments, and anglesto solve realworld type problems. In ClassVIII the focus is on:
- Finding the distance between two points.
- Finding the midpoint of a line segment.
- Finding the slope of a line.
- Writing the equation of a line in the form y = mx + c or ax + by + c = 0.
- Understanding the relationship between parallelism, perpendicularity, and slopes.
Key Concepts You Must Master
1. Distance Formula
For points P(x,y) and Q(x,y), the distance PQ is:
\( \displaystyle PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
2. Midpoint Formula
The midpoint M of PQ has coordinates:
\( \displaystyle M\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right) \)
3. Slope of a Line
The slope m of the line through P(x,y) and Q(x,y) is:
\( \displaystyle m = \frac{y_2-y_1}{x_2-x_1} \)
A zero slope means a horizontal line; an undefined (or infinite) slope means a vertical line.
4. Equation of a Line
If the slope m and a point (x,y) are known, the line can be written as:
\( \displaystyle y-y_1 = m(x-x_1) \) (pointslope form)
When the equation must be in standard form, expand and rearrange to obtain ax + by + c = 0.
5. Parallel & Perpendicular Lines
- Parallel lines have equal slopes (m = m).
- Perpendicular lines satisfy mm = -1 (product of slopes is 1).
StepbyStep Solution Strategy for Exercise4.4
- Read the question carefully. Identify which quantities are given (coordinates, slope, parallelism, etc.) and what is being asked for.
- Translate words into mathematical expressions. Draw a diagram whenever possible; label all known points.
- Choose the appropriate formula. Use distance for length, midpoint for centre, slope for gradient, and the lineequation formula for the required equation.
- Substitute the known values. Keep the algebra neat; simplify fractions early to avoid mistakes.
- Check your answer. Substitute the answer back into the original condition (e.g., verify that the line passes through the offered point).
WorkedOut Examples from Exercise4.4
Example1 Finding Distance
Question: Find the distance between the points A(2,3) and B(7,1).
Solution:
Use the distance formula:
\( AB = \sqrt{(7-2)^2 + (-1-3)^2} = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.40 \text{ units} \)
The distance between A and B is \( \sqrt{41} \) units (6.40).
Example2 Midpoint of a Segment
Question: Determine the midpoint of the line segment joining P(2,5) and Q(4,3).
Solution:
Midpoint M is given by:
\( M\left(\frac{-2+4}{2},\;\frac{5+(-3)}{2}\right) = M\left(\frac{2}{2},\;\frac{2}{2}\right) = M(1,1) \)
Hence, the midpoint is the point (1,1).
Example3 Equation of a Line Through Two Points
Question: Write the equation of the line passing through C(1,2) and D(5,6).
Solution:
- Calculate slope:
\( m = \frac{-6-2}{5-1} = \frac{-8}{4} = -2 \) - Use pointslope form with point C(1,2):
\( y-2 = -2(x-1) \) - Expand and rearrange to standard form:
\( y-2 = -2x + 2 \)
\( 2x + y - 4 = 0 \)
The required line is 2x + y 4 = 0.
Example4 Parallel & Perpendicular Lines
Question: The line L has equation 3x 4y + 5 = 0. Find the equation of a line L that is (a) parallel to L and passes through the point (2,3); (b) perpendicular to L and passes through the same point.
Solution (a) Parallel line:
- Rewrite L in slopeintercept form:
\( 3x - 4y + 5 = 0 -4y = -3x -5 y = \frac{3}{4}x + \frac{5}{4} \) The slope \(m = \frac{3}{4}\). - A parallel line has the same slope \(m = \frac{3}{4}\). Using point (2,3):
\( y-3 = \frac{3}{4}(x-2) \) - Convert to standard form:
\(4(y-3) = 3(x-2) 4y - 12 = 3x - 6\) \(3x - 4y + 6 = 0\) (or any equivalent multiple).
Solution (b) Perpendicular line:
- Perpendicular slope is the negative reciprocal: \( m_{\perp} = -\dfrac{4}{3} \).
- Using point (2,3):
\( y-3 = -\frac{4}{3}(x-2) \) - Standard form:
\(3(y-3) = -4(x-2) 3y - 9 = -4x + 8\) \(4x + 3y - 17 = 0\).
Thus, the required equations are 3x 4y + 6 = 0 (parallel) and 4x + 3y 17 = 0 (perpendicular).
Common Mistakes to Avoid
- Sign errors in the distance formula. Remember to square the differences; the sign disappears after squaring.
- Swapping coordinates in the midpoint formula. Keep the order (xcoordinates together, ycoordinates together).
- Dividing by zero when finding slope. If \(x_2 = x_1\), the line is vertical and the slope is undefined.
- Forgetting to check the condition of parallelism/perpendicularity. Always verify that the product of slopes is 1 for perpendicular lines.
Quick Revision Checklist
- Write down the given coordinates correctly.
- Identify the formula needed (distance, midpoint, slope, equation).
- Calculate slope first; it often simplifies later steps.
- After obtaining the equation, substitute the given point(s) to confirm.
- Round off only at the final step if a decimal answer is required.
Tip for exams: Sketch a quick diagram even for algebraic problems. Visualising the points helps you to avoid sign mistakes and to decide which formula to apply.
Additional Practice Questions
Try these on your own to gain confidence:
- Find the distance between (3,4) and (5,2).
- Determine the midpoint of the segment joining (7,0) and (1,8).
- Write the equation of the line that passes through (2,5) and has a slope of 3/2.
- Two lines are perpendicular. One line has the equation y = x + 7. Write the equation of the line passing through the point (4,1) that is perpendicular to it.
- A line L passes through (3,2) and (7,k). If L is parallel to the line 2x 3y + 5 = 0, find the value of k.
Further Reading & Resources
For deeper insight, explore the following NCERT chapters and external resources:
- NCERT ClassVIII Mathematics Chapter4: Practical Geometry (full textbook PDF).
- Khan Academy Coordinate Geometry playlist (covers distance, midpoint, and slope).
- Byjus CBSE ClassVIII Geometry Solutions (video explanations of Exercise4.4).
- National Digital Library of India Interactive geometry applets.
Conclusion
Exercise4.4 is designed to test your ability to apply fundamental geometric formulas to realworld style problems. By mastering the distance, midpoint, and slope concepts and by practising the systematic approach outlined above, you will be able to solve all the questions with confidence. Remember to keep a tidy notebook, draw clear diagrams, and doublecheck every arithmetic step. Happy studying!
For any doubts, feel free to contact your mathematics teacher or join an online discussion forum where classmates share solution strategies.
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