Class 10 Maths Coordinate Geometry Notes

7.1 Introduction to Coordinate Geometry

Coordinate Geometry is the branch of geometry that uses a coordinate system to study geometric shapes and figures. It combines algebra and geometry to find the positions of points, lines, and other figures in a plane using coordinates.

The Cartesian Coordinate System, introduced by René Descartes, is a two-dimensional system in which the position of a point is represented by an ordered pair $(x, y)$(x,y), where:

• $x$x is the horizontal distance from the origin (x-axis),
• $y$y is the vertical distance from the origin (y-axis).

In this chapter, we will explore some of the key concepts and formulas related to coordinate geometry, including:

• Distance Formula: To calculate the distance between two points in the plane.
• Section Formula: To find the coordinates of a point that divides a line segment in a given ratio.

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7.2 Distance Formula

The Distance Formula helps us calculate the distance between two points $(x_1, y_1)$(x1​,y1​) and $(x_2, y_2)$(x2​,y2​) in a coordinate plane.

The distance $d$d between two points is given by the formula:$$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$$d=(x2​−x1​)2+(y2​−y1​)2​

This formula is derived from the Pythagorean Theorem. The difference in the x-coordinates $(x_2 – x_1)$(x2​−x1​) and the difference in the y-coordinates $(y_2 – y_1)$(y2​−y1​) form the two sides of a right-angled triangle, and the distance is the hypotenuse.

Example:

Find the distance between the points $A(1, 2)$A(1,2) and $B(4, 6)$B(4,6).

Using the distance formula:$$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$$d=(x2​−x1​)2+(y2​−y1​)2​

Substitute the coordinates of $A(1, 2)$A(1,2) and $B(4, 6)$B(4,6):$$d = \sqrt{(4 – 1)^2 + (6 – 2)^2}$$d=(4−1)2+(6−2)2​ $$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$d=32+42​=9+16​=25​=5

So, the distance between the points $A(1, 2)$A(1,2) and $B(4, 6)$B(4,6) is 5 units.

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7.3 Section Formula

The Section Formula is used to find the coordinates of a point that divides a line segment joining two points in a given ratio. This point can be inside or outside the segment depending on the ratio.

The formula for finding the coordinates of the point $P(x, y)$P(x,y) dividing the line segment joining $A(x_1, y_1)$A(x1​,y1​) and $B(x_2, y_2)$B(x2​,y2​) in the ratio $m : n$m:n is:$$x = \frac{mx_2 + nx_1}{m + n}$$x=m+nmx2​+nx1​​ $$y = \frac{my_2 + ny_1}{m + n}$$y=m+nmy2​+ny1​​

Where:

• $A(x_1, y_1)$A(x1​,y1​) and $B(x_2, y_2)$B(x2​,y2​) are the coordinates of the two points.
• $m : n$m:n is the ratio in which point $P$P divides the segment $AB$AB.

Example:

Find the coordinates of the point $P$P that divides the line segment joining $A(2, 3)$A(2,3) and $B(5, 7)$B(5,7) in the ratio $2 : 3$2:3.

Using the Section Formula:$$x = \frac{2 \cdot 5 + 3 \cdot 2}{2 + 3} = \frac{10 + 6}{5} = \frac{16}{5} = 3.2$$x=2+32⋅5+3⋅2​=510+6​=516​=3.2 $$y = \frac{2 \cdot 7 + 3 \cdot 3}{2 + 3} = \frac{14 + 9}{5} = \frac{23}{5} = 4.6$$y=2+32⋅7+3⋅3​=514+9​=523​=4.6

So, the coordinates of point $P$P are $(3.2, 4.6)$(3.2,4.6).

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7.4 Summary

In this chapter, we have covered key concepts related to Coordinate Geometry:

1. Coordinate System: The Cartesian coordinate system helps locate points on a plane using ordered pairs $(x, y)$(x,y).
2. Distance Formula: The formula $d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$d=(x2​−x1​)2+(y2​−y1​)2​ helps calculate the distance between two points in the plane.
3. Section Formula: The section formula allows us to find the coordinates of a point dividing a line segment in a given ratio.

These concepts are essential for solving geometric problems involving distance and division of line segments, and they form the foundation of more advanced topics in geometry.

MCQs Based on the “Coordinate Geometry” Chapter:

1. What is the distance between the points A(3,4)A(3, 4)A(3,4) and B(7,1)B(7, 1)B(7,1)?

a) $\sqrt{25}$25​
b) $\sqrt{20}$20​
c) $\sqrt{13}$13​
d) $\sqrt{17}$17​

Answer: a) $\sqrt{25}$25​

2. The coordinates of the midpoint of the line segment joining A(x1,y1)A(x_1, y_1)A(x1​,y1​) and B(x2,y2)B(x_2, y_2)B(x2​,y2​) are:

a) $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$(2×1​+x2​​,2y1​+y2​​)
b) $\left( x_1 + x_2, y_1 + y_2 \right)$(x1​+x2​,y1​+y2​)
c) $\left( \frac{x_1 – x_2}{2}, \frac{y_1 – y_2}{2} \right)$(2×1​−x2​​,2y1​−y2​​)
d) $\left( x_1 – x_2, y_1 – y_2 \right)$(x1​−x2​,y1​−y2​)

Answer: a) $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$(2×1​+x2​​,2y1​+y2​​)

3. Find the coordinates of the point that divides the segment joining A(1,2)A(1, 2)A(1,2) and B(3,8)B(3, 8)B(3,8) in the ratio 1:21:21:2.

a) $(2, 4)$(2,4)
b) $(2.5, 5)$(2.5,5)
c) $(2, 6)$(2,6)
d) $(1.5, 4)$(1.5,4)

Answer: b) $(2.5, 5)$(2.5,5)

4. The distance between the points A(2,3)A(2, 3)A(2,3) and B(2,7)B(2, 7)B(2,7) is:

a) 4
b) 5
c) 3
d) 6

Answer: a) 4