CBSE Class 10 Mathematics

Why Coordinate Geometry Matters

Coordinate Geometry links algebra with geometry, allowing us to solve geometric problems using algebraic methods. In Class10, this chapter lays the groundwork for handling points, lines, distances, and midpoints on the Cartesian plane concepts that reappear in higher classes and various competitive exams.

What the Chapter Covers

Chapter7 introduces the following key ideas:

• Plotting points on the Cartesian plane.
• Distance formula between two points.
• Midpoint formula.
• Section formula (internal division).
• Collinearity and slope.

Exercises 7.17.5 develop problemsolving skills based on these concepts. Exercise7.1 specifically focuses on the distance, midpoint, and section formulas.

Exercise 7.1 Overview

Exercise7.1 contains 10 questions that ask the student to:

1. Find the distance between given points.
2. Determine the coordinates of the midpoint of a segment.
3. Locate a point that divides a line segment in a given ratio.
4. Use the results to answer related geometric questions such as checking collinearity.

These questions test basic manipulation of the formulas and require clear, stepbystep working.

Key Formulas to Remember

Distance between (x , y) and (x , y):
d = [(x x) + (y y)]

Midpoint of (x , y) and (x , y):
M = ( (x + x)/2 , (y + y)/2 )

Section formula (internal division) point dividing (x , y) and (x , y) in ratio m:n:
P = ( (mx + nx)/(m+n) , (my + ny)/(m+n) )

Always simplify fractions and square roots wherever possible. Check your answer by plugging the point back into the distance formula if required.

StepbyStep Solutions for Common Types of Questions

All 10 questions of Exercise7.1 can be solved using the same systematic approach: write down known coordinates, substitute them into the appropriate formula, simplify, and finally verify the answer.

Quick Tips for Solving Exercise7.1

• Mind the signs: A common mistake is to overlook negative coordinates when squaring differences.
• Keep fractions rational: If the ratio m:n is not reduced, simplify it first to avoid large numbers.
• Check your work: After obtaining a point, substitute it back into the distance formula with the other endpoint; the result should match the required distance (if the problem asks for it).
• Use symmetry: For midpoint problems, the coordinates are simply the arithmetic averages, which often help verify the answer instantly.
• Draw a sketch: A quick sketch of the points on graph paper helps visualise the geometry and prevents sign errors.

Sample Full Solution for Question5 (From Exercise7.1)

Question: Find the distance between the points P(4,7) and Q(2,1). Then determine the midpoint of PQ.

Solution:

1. Distance:
   d = [(2(4)) + (17)] = [(6) + (8)] = [36 + 64] = 100 = 10.
2. Midpoint:
   M = ( (4+2)/2 , (7+(1))/2 ) = ( (2)/2 , (6)/2 ) = (1 , 3).
3. Result: Distance = 10 units, Midpoint = (1,3).

Notice how the distance is a whole number because the squared differences summed to a perfect square, a useful check after computation.

Frequently Asked Questions (FAQs)

1. What if the ratio in a section problem is given as 1:0?

That means the point coincides with the first endpoint; the coordinates are exactly those of the first point.

2. How do I know whether to use the internal or external section formula?

Exercise7.1 deals only with internal division. An external division would have a negative ratio and is not required for this set.

3. Can the distance formula be used for threedimensional space?

Yes, but the formula extends to d = [(xx) + (yy) + (zz)]. In Class10 we only consider the 2D plane.

4. Why does the answer sometimes simplify to an integer?

When the squares of the differences add up to a perfect square, the square root simplifies to an integer. This is a good sanity check.

Conclusion

Exercise7.1 of Chapter7 reinforces the essential formulas that form the backbone of Coordinate Geometry. By practising each type of problemdistance, midpoint, and sectionstudents develop a solid foundation that will be valuable for later topics such as equations of lines and circles. Consistent practice, careful handling of signs, and verification through substitution are the keys to mastering these problems.

For more detailed solutions of the remaining questions, students can refer to the official NCERT solution book or credible online platforms that provide stepbystep explanations aligned with the CBSE syllabus.