NCERT Solutions for Class 10 Maths Chapter 7 - Coordinate Geometry

Introduction to Coordinate Geometry

Coordinate Geometry, also known as Analytic Geometry, is a branch of mathematics that combines geometry with algebra to study geometric problems using coordinate systems. This chapter introduces students to the concept of representing points, lines, and shapes using a coordinate system, which forms the foundation for advanced mathematical concepts in higher education.

Overview of Chapter 7

NCERT Class 10 Maths Chapter 7 "Coordinate Geometry" is designed to help students understand the relationship between geometry and algebra. The chapter covers various important formulas and methods to solve problems related to points in a coordinate plane. It builds upon the basic knowledge of coordinate geometry from previous classes and introduces more advanced concepts.

The chapter is divided into four main exercises, each focusing on different aspects of coordinate geometry. By the end of this chapter, students will be able to calculate distances between two points, find the coordinates of a point dividing a line segment in a given ratio, and determine the area of triangles formed by three points in a coordinate plane.

Key Concepts and Formulas

1. Distance Formula

The distance between two points A(x, y) and B(x, y) in a coordinate plane is given by:

This formula is derived from the Pythagorean theorem and is essential for calculating the length of line segments in a coordinate plane.

2. Section Formula

If point P(x, y) divides the line segment joining A(x, y) and B(x, y) in the ratio m:n, then the coordinates of P are:

When point P is the midpoint of AB (i.e., divides AB in the ratio 1:1), the formula simplifies to:

3. Area of a Triangle

The area of a triangle with vertices A(x, y), B(x, y), and C(x, y) is given by:

Three points are collinear (lie on the same line) if the area of the triangle formed by them is zero.

Exercise-wise Solutions

Exercise 7.1 - Distance Formula

Exercise 7.1 focuses on problems related to the distance formula. Students learn to calculate the distance between two points and apply this concept to solve various geometric problems.

Exercise 7.2 - Section Formula

Exercise 7.2 deals with the section formula, which is used to find the coordinates of a point that divides a line segment in a given ratio. This exercise includes problems on finding the midpoint and other points dividing a line segment.

Exercise 7.3 - Section Formula

This exercise extends the application of the section formula to more complex problems, including finding coordinates in various scenarios such as finding the ratio in which a point divides a line segment.

Exercise 7.4 - Area of a Triangle

Exercise 7.4 focuses on problems related to calculating the area of triangles formed by points in a coordinate plane. It also includes problems on identifying collinear points based on the area of the triangle formed by them.

Important Questions and Solutions

Question 1:

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

Solution:

In a parallelogram, opposite sides are parallel and equal. So, the midpoints of the diagonals are the same.

Midpoint of diagonal joining (1, 2) and (x, 6) = Midpoint of diagonal joining (4, y) and (3, 5)

Comparing the coordinates:

Therefore, x = 6 and y = 3.

Question 2:

The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices.

Solution:

Let the opposite vertices of the square be A(-1, 2) and C(3, 2).

The side length of the square is the distance between A and C:

The diagonal of the square is 4 units. The midpoint of AC is:

The other diagonal is perpendicular to AC and has the same length. Let the other vertices be B(x, y) and D(x', y').

Since B and D are vertices of the square, the distance between B and the midpoint is half the diagonal, which is 2 units.

Also, B is at a 90 angle from A, so the slope of AB is undefined (vertical line) because AC is horizontal.

Therefore, B has x-coordinate -1, and to be at a distance of 2 units from the midpoint (1, 2):

Similarly, D has x-coordinate 3:

Since AC = 4 and the side length is 4/2 = 22, we check the distance between A and B:

For B(-1, 0): AB = [(-1 - (-1)) + (0 - 2)] = [0 + 4] = 2 (This is not equal to 22)

For B(-1, 4): AB = [(-1 - (-1)) + (4 - 2)] = [0 + 4] = 2 (This is not equal to 22)

There seems to be an error in our approach. Let me reconsider.

The side length of the square is the distance between adjacent vertices, not opposite vertices. The diagonal length is 4 units.

The side length (s) of the square is diagonal/2 = 4/2 = 22

The midpoint of AC is (1, 2). The other diagonal BD passes through this midpoint and is perpendicular to AC.

Since AC is horizontal (its endpoints have the same y-coordinate), BD is vertical.

The length of BD is also 4 units (since diagonals of a square are equal).

So, the endpoints of BD are 2 units above and below the midpoint (1, 2).

Therefore, the other two vertices of the square are (1, 4) and (1, 0).

Tips for Exam Preparation

• Practice all the exercises from the NCERT textbook thoroughly.
• Memorize the important formulas: distance formula, section formula, and area of a triangle formula.
• Understand the derivation of these formulas to gain a deeper insight into the concepts.
• Solve additional problems from reference books to strengthen your understanding.
• Pay attention to special cases, such as when points are collinear or when a point divides a line segment externally.
• Draw diagrams to visualize the problems more clearly.
• Learn to identify which formula is appropriate for different types of problems.
• Practice time management by solving problems within a time limit.
• Review your mistakes and learn from them.
• Attempt previous years' question papers to understand the pattern of questions.

Conclusion

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry provides students with a comprehensive understanding of how geometry and algebra work together. Through the distance formula, section formula, and area of a triangle formula, students learn to solve various geometric problems using algebraic methods.

By mastering these concepts, students develop analytical thinking skills and problem-solving abilities that are essential for higher studies in mathematics and science. Regular practice and application of these formulas enable students to approach coordinate geometry problems with confidence and accuracy.

The chapter not only prepares students for their Class 10 examinations but also lays the groundwork for more advanced mathematical concepts they will encounter in future studies. Understanding coordinate geometry is crucial for fields such as physics, engineering, computer graphics, and various other technical disciplines.

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