Class 10 Mathematics • Coordinate Geometry • Chapter 7
Chapter Focus:
Coordinate Geometry connects algebra with geometry
by representing points and line segments using ordered
pairs of numbers.
The most important ideas in this chapter are:
1.
Understanding the coordinate plane and ordered pairs
2.
Finding the distance between two points
3.
Finding the coordinates of a point dividing a
line segment in a given ratio
4.
Applying the formulas to geometric problems
A coordinate plane is formed by two perpendicular
number lines.
The horizontal number line is called the
x-axis, while the vertical number line
is called the y-axis.
The point where the two axes intersect is called the
origin and is represented by
(0, 0).
Cartesian Coordinate Plane
Important:
Quadrant I → (+,+)
Quadrant II → (−,+)
Quadrant III → (−,−)
Quadrant IV → (+,−)
Abscissa
The x-coordinate of a point.
Ordinate
The y-coordinate of a point.
Origin
The point (0, 0).
Ordered Pair
A point written as (x, y).
Remember:
The first coordinate always gives the horizontal
position and the second coordinate gives the vertical
position.
7.2 Distance Formula
Consider two points
A(x₁, y₁)
and
B(x₂, y₂)
.
Draw perpendiculars from A and B to the coordinate
axes. These perpendicular segments form a right
triangle.
The horizontal side has length
|x₂ − x₁|
and the vertical side has length
|y₂ − y₁|
.
Distance Between Two Points
The distance AB is the hypotenuse of the
right triangle. Therefore the Pythagorean theorem
gives the distance formula.
Distance AB
=
√[(x₂ − x₁)² + (y₂ − y₁)²]
Derivation
Let the horizontal difference be
x₂ − x₁
and the vertical difference be
y₂ − y₁
.
Common Mistake:
Do not forget to square both coordinate differences.
The signs disappear after squaring.
📌 Special Cases of Distance
Same x-coordinate
If x₁ = x₂, the line is vertical.
Distance = |y₂ − y₁|
Same y-coordinate
If y₁ = y₂, the line is horizontal.
Distance = |x₂ − x₁|
Distance from Origin
For P(x,y):
OP = √(x² + y²)
7.3 Section Formula
Suppose a point P divides the line segment joining
A(x₁,y₁)
and
B(x₂,y₂)
internally in the ratio
m : n
.
Internal Division in the Ratio m : n
P lies between A and B and divides AB internally
in the ratio m:n.
P =
(
mx₂ + nx₁
m + n
,
my₂ + ny₁
m + n
)
How to Remember the Formula
Multiply the coordinate of B by
m
and the coordinate of A by
n
.
Add them and divide by
m+n
.
Example 2
Find the coordinates of the point P that divides
A(2,3) and B(8,9) internally in the ratio 2:1.
P =
(
2×8 + 1×2
2+1
,
2×9 + 1×3
2+1
)
= (6,7)
Therefore,
P = (6,7)
.
📍 Midpoint Formula
The midpoint is the point that divides a line segment
into two equal parts.
Midpoint =
(
x₁ + x₂
2
,
y₁ + y₂
2
)
Midpoint of a Line Segment
Example 3
Find the midpoint of
A(−4,2)
and
B(6,8)
.
M =
(
−4 + 6
2
,
2 + 8
2
)
= (1,5)
Therefore,
M = (1,5)
.
🎯 Applications of Coordinate Geometry
1. Checking Whether a Triangle is Isosceles
Find all three side lengths using the Distance Formula.
If two sides are equal, the triangle is isosceles.
2. Checking Whether a Triangle is Equilateral
Calculate all three side lengths.
If all three distances are equal, the triangle
is equilateral.
3. Checking Right-Angled Triangle
Calculate the squares of the three side lengths.
If
a² + b² = c²
for the largest side c, then the triangle is
right-angled.
4. Finding a Point Dividing a Segment
Use the Section Formula when a ratio is given.
For equal division, use the Midpoint Formula.
5. Finding Missing Coordinates
In many coordinate geometry problems, a distance,
midpoint or ratio is given and one coordinate is unknown.
Substitute the known values into the appropriate formula
and solve the resulting equation.
Problem-Solving Strategy:
Step 1 → Write the coordinates clearly.
Step 2 → Select the correct formula.
Step 3 → Substitute carefully.
Step 4 → Simplify step by step.
Step 5 → Check whether the answer is reasonable.