Short Notes: Coordinate Geometry

# Coordinate Geometry Class 10 Notes Maths Chapter 7

 Table of contents Terminologies related to Cartesian Plane Distance Formula Section Formula Finding Ratios given the Points Mid Point of a Line Segment Points of Trisection Centroid of a Triangle

• Position of a point P in the Cartesian plane with respect to co-ordinate axes is represented by the ordered pair (x, y).

• The line X’OX is called the X-axis and YOY’ is called the Y-axis.
• The part of intersection of the X-axis and Y-axis is called the origin O and the co-ordinates of O are (0, 0).
• The perpendicular distance of a point P from the Y-axis is the ‘x’ co-ordinate and is called the abscissa.
• The perpendicular distance of a point P from the X-axis is the ‘y’ co-ordinate and is called the ordinate.
• Signs of abscissa and ordinate in different quadrants are as given in the diagram:

• Any point on the X-axis is of the form (x, 0).
• Any point on the Y-axis is of the form (0, y).

Question for Short Notes: Coordinate Geometry
Try yourself:If a point P in the Cartesian plane has coordinates (5, -3), in which quadrant does this point lie, and what are its abscissa and ordinate values?

## Distance Formula

The distance between two points P(x1, y1) and Q (x2, y2) is given by

Note. If O is the origin, the distance of a point P(x, y) from the origin O(0, 0) is given by :

Example: Find the distance between the following points:
(i) (-1, 2) and (2, 3)
(ii) (0, 1) and (6, –1)

Solution:

(i) Let the distance between the points (-1, 2) and (2, 3) be d, then

d = √[(2 – ( –1))2 + (3 – 2)2] = √[9 + 1] = √10 units.

(ii) Let the distance between the points (0, 1) and (6, –1) be d, then

d = √[(6 – 0)2 + ( –1 – 1)2] = √[36 + 4] = √40 = 2√10 units.

## Section Formula

The coordinates of the point which divides the line segment joining the points A(x1, y1) and B(x2, y2) internally in the ratio m : n are:

The above formula is section formula.

The ratio m: n can also be written as m/n  : 1 or k:1, The co-ordinates of P can also be written as P(x, y) =

Example: Find the coordinates of the point which divides the line segment joining the points (4,6) and (-5,-4) internally in the ratio 3:2.

Sol: Let P(x, y) be the point which divides the line segment joining A(4, 6) and B(-5, -4) internally in the ratio 3 : 2.

Here,

(x1, y1) = (4, 6)

(x2, y2) = (-5, -4)

m : n = 3 : 2

Using the section formula,

Coordinates of P are,

Therefore, P(x,y) = (-7/5,0).

## Finding Ratios given the Points

To find the ratio in which a given point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2),

• Assume that the ratio is k : 1
• Substitute the ratio in the section formula for any of the coordinates to get the value of k.

When x1, x2 and x are known, k can be calculated. The same can be calculated from the y- coordinate also.

Example: Find the ratio when point (– 4, 6) divide the line segment joining the points A(– 6, 10) and B(3, – 8)?

Solution: Let the ratio be m:n.

We can write the ratio as:

m/n : 1 or k:1

Suppose (-4, 6) divide the line segment AB in k:1 ratio.

Now using the section formula, we have the following;

Thus, the required ratio is 2:7.

## Mid Point of a Line Segment

The mid-point of the line segment joining the points P(x1, y1) and Q(x2, y2) is

Substituting m = 1, n = 1 in section formula we get,

Question for Short Notes: Coordinate Geometry
Try yourself:Given two points A(3, 4) and B(7, 1) in the Cartesian plane, find the distance between these two points.

## Points of Trisection

To find the points of trisection P and Q, which divides the line segment joining A(x1, y1) and B(x2, y2) into three equal parts:

i) AP : PB = 1 : 2

ii) AQ : QB = 2 : 1

Example: Find the coordinates of the points of trisection of the line segment joining the points A(2, – 2) and B(– 7, 4).

Solution: Let P and Q divide the line segment AB into three parts.

So, P and Q are the points of trisection here.

Let P divides AB in 1:2, thus by section formula, the coordinates of P are (1, 0)

Let Q divides AB in 2:1 ratio, then by section formula, the coordinates are (-4,2)

Thus, the point of trisection for line segment AB are (1,0) and (-4,2).

## Centroid of a Triangle

If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of a ΔABC, then the coordinates of its centroid(P) are given by:

Example: Find the coordinates of the centroid of a triangle whose vertices are given as (-1, -3), (2, 1) and (8, -4)

Solution: Given,

The coordinates of the vertices of a triangle are (-1, -3), (2, 1) and (8, -4)

The Centroid of a triangle is given by:

G = ((x1+x2+x3)/3, (y1+y2+y3)/3)

G = ((-1+2+8)/3, (-3+1-4)/3)

G = (9/3, -6/3)

G = (3, -2)

Therefore, the centroid of a triangle, G = (3, -2)

The document Coordinate Geometry Class 10 Notes Maths Chapter 7 is a part of the Class 10 Course Mathematics (Maths) Class 10.
All you need of Class 10 at this link: Class 10

## Mathematics (Maths) Class 10

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## FAQs on Coordinate Geometry Class 10 Notes Maths Chapter 7

 1. What is the Distance Formula in the Cartesian Plane?
Ans. The Distance Formula is used to determine the distance between two points in the Cartesian Plane. If you have two points, $$P(x_1, y_1)$$ and $$Q(x_2, y_2)$$, the distance $$d$$ between them is given by the formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
 2. How do you find the Midpoint of a Line Segment?
Ans. The Midpoint of a line segment that connects two points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$ is calculated by averaging the x-coordinates and the y-coordinates of the two points. The formula is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
 3. What is the Section Formula and how is it used?
Ans. The Section Formula is used to find the coordinates of a point that divides a line segment into a particular ratio. If point $$P$$ divides the segment joining points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$ in the ratio $$m:n$$, the coordinates of $$P$$ are given by: $P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$
 4. How can you find the Points of Trisection of a Line Segment?
Ans. To find the Points of Trisection of a line segment joining points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$, you divide the segment into three equal parts. The coordinates of the points of trisection, $$T_1$$ and $$T_2$$, can be calculated using the formulas: $T_1 = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right)$ $T_2 = \left( \frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3} \right)$
 5. What is the Centroid of a Triangle and how is it calculated?
Ans. The Centroid of a Triangle is the point where the three medians intersect and it serves as the triangle's center of mass. If the vertices of the triangle are $$A(x_1, y_1)$$, $$B(x_2, y_2)$$, and $$C(x_3, y_3)$$, the coordinates of the centroid $$G$$ can be calculated using the formula: $G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$

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