Coordinate Geometry basic formulas

1. Coordinate Geometry
Concept Session

2. The Cartesian plane
2

3. Distance between 2 Points P(x1, y1) and Q(x2, y2)
Distance PQ :
𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
Distance from origin of point P :
𝑥1
2 + 𝑦1
2
3

4. Distance between 2 Points - Problems
Let the vertices of a triangle ABC be (−8, 8), (2, −8) and (−2, 2) then the triangle is
1. Right angled
2. Equilateral
3. Isosceles
4. Scalene
4

5. The midpoint of an interval
The midpoint of an interval with endpoints 𝑃(𝑥1, 𝑦1) and 𝑄(𝑥2, 𝑦2) is
𝑥1 + 𝑥2
2
,
𝑦1 + 𝑦2
2
5

6. The midpoint of an interval - Problem
If C(3, 5) is the midpoint of line interval AB and A has coordinates (–2, 1), find the
coordinates of B.
6
Answer : (4, 9)

7. Section formula – Internal division
Given two end points of line segment A(x1, y1) and B (x2, y2)
you can determine the coordinates of the point P(x, y)
that divides the given line segment in the ratio m : n
internally using Section Formula
𝑥 =
𝑚𝑥2+𝑛𝑥1
𝑚+𝑛
𝑦 =
𝑚𝑦2+𝑛𝑦1
𝑚+𝑛
7

8. Section formula – External division
Given two end points of line segment A(x1, y1) and B (x2, y2)
you can determine the coordinates of the point P(x, y)
that divides the given line segment in the ratio m : n
externally using Section Formula given by
𝑥 =
𝑚𝑥2−𝑛𝑥1
𝑚−𝑛
𝑦 =
𝑚𝑦2−𝑛𝑦1
𝑚−𝑛
8

9. Section formula – Problem
In what ratio is the line joining (2, 3) and (3, 7)
divided by 3x + y = 9?
9
Answer: Externally in the
ratio 2 : -7

10. Area of a Triangle
The area of a triangle ABC whose vertices are A(x1, y1), B
(x2, y2) and C(x3, y3) is denoted by
1
2
| 𝑥1 𝑦2 − 𝑥2 𝑦1 + 𝑥2 𝑦3 − 𝑥3 𝑦2 + 𝑥3 𝑦1 − 𝑥1 𝑦3 |
10

11. Area of a Polygon
The area of a polygon whose vertices are (𝑥1, 𝑦1),
(𝑥2, 𝑦2) and (𝑥3, 𝑦3) … 𝑥 𝑛, 𝑦𝑛 is denoted by
1
2
| 𝑥1 𝑦2 − 𝑥2 𝑦1 + 𝑥2 𝑦3 − 𝑥3 𝑦2 + 𝑥3 𝑦1 − 𝑥1 𝑦3 … 𝑥 𝑛 𝑦1 − 𝑥1 𝑦𝑛 |
11

12. Centroid of a triangle
Centroid – meeting point of the three medians.
The centroid(G) of a triangle ABC whose vertices
are A(𝑥1, 𝑦1), B(𝑥2, 𝑦2) and C(𝑥3, 𝑦3) is denoted by
𝐺 =
𝑥1 + 𝑥2 + 𝑥3
3
,
𝑦1 + 𝑦2 + 𝑦3
3
12

13. Centroid of a triangle - Problem
If (−3,2) is the centroid of a triangle two of whose
vertices are (3, −4) and −5, 2 . Find the area of the
triangle.
1. 27 units
2. 18 units
3. 36 units
4. None of these
13
Ans : 18 units

14. Incenter of a triangle
If 𝐴(𝑥1, 𝑦1), 𝐵 (𝑥2, 𝑦2) and 𝐶(𝑥3, 𝑦3) are the
vertices of the triangle ABC such that
BC=a, CA=B and AB=c, then the
coordinates of its incentre are
𝐴(𝑥1, 𝑦1)
𝐵(𝑥2, 𝑦2) 𝐶(𝑥3, 𝑦3)𝑎
𝑐 𝑏𝐼 =
𝑎𝑥1 + 𝑏𝑥2 + 𝑐𝑥3
𝑎 + 𝑏 + 𝑐
,
𝑎𝑦1 + 𝑏𝑦2 + 𝑐𝑦3
𝑎 + 𝑏 + 𝑐
14

15. Straight Line
▪ Slope of a Line
▪ Equations of a
straight line
– General form of
Equation
– Slope-intercept
form
– Point slope form
– Two point form
– Intercept form
– Normal form
▪ Angle between two
straight lines
▪ Distance of a point from a
line
▪ Distance between two
parallel lines
15

16. Slope of a line ( Gradient)
The slope, represented by the letter m, measures the
inclination or steepness of the line.
The slope is always measured anti-clock wise
𝑆𝑙𝑜𝑝𝑒 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
=
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
𝑆𝑙𝑜𝑝𝑒 =
∆𝑦
∆𝑥
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
16

17. Slope of a line ( Gradient) - values
Positive Slope – On moving from left
to right, the line rises
Negative Slope - On moving from left
to right, the line dips
17

18. Slope of a line ( Gradient) - values
Zero Slope – Parallel to x axis and
hence “No rise”
Undefined Slope – Parallel to y axis and
hence “ infinite rise”
18

19. Slope of a special line
Parallel Lines  equal slope
𝑚1 = 𝑚2
Perpendicular lines  negative inverses
𝑚1 ∗ 𝑚2 = −1
19

20. Equation of a straight line – General Form
𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0
Slope of the line =
−𝑎
𝑏
Y intercept =
𝑐
𝑏
20

21. Equation of a straight line – Slope intercept
Form
Equation of a straight line whose slope is 𝑚 and
which cuts of a y-intercept of 𝐶 units is given by
𝑦 = 𝑚𝑥 + 𝐶
21

22. Equation of a straight line – Problems
Find the equation of a line that intersects the Y
axis at the point (0,3) and has a slope of
5
3
Answer : 3𝑦 − 5𝑥 = 9
22

23. Equation of a straight line – Point-slope Form
Equation of a straight line passing through a point
(𝑥1, 𝑦1) and having a slope m is given by
𝑦 − 𝑦1 = 𝑚 ∗ (𝑥 − 𝑥1)
23

24. Equation of a straight line – Problems
Find an equation of the line that passes through (4, 6) and
is parallel to the line whose equation is 𝑦 =
2
3
𝑥 + 5.
1. 2𝑦 = 3𝑥 + 10
2. 3𝑦 = 3𝑥 + 10
3. 3𝑦 = 2𝑥 + 10
4. 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
Answer: Option 3
24

25. Equation of a straight line – Problems
Find an equation of the line that passes through (4, 6) and
is perpendicular to the line whose equation is 𝑦 =
2
3
𝑥 + 5.
1. 2𝑦 = −3𝑥 + 10
2. 3𝑦 = −3𝑥 + 24
3. 3𝑦 = 2𝑥 + 10
4. 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
Answer: 2𝑦 = −3𝑥 +
24
25

26. Equation of a straight line – Two-point Form
Equation of a straight line passing through two points
(𝑥1, 𝑦1) and (𝑥2, 𝑦2) is given by
𝑦 − 𝑦1 =
(𝑦2−𝑦1)
(𝑥2−𝑥1)
∗ (𝑥 − 𝑥1)
26

27. Equation of a straight line – Intercept Form
Equation of a straight line which cuts off x-intercept (a)
and y-intercept (b) is
𝑥
𝑎
+
𝑦
𝑏
= 1
27

28. Equation of a straight line – Normal Form
If 𝑝 is the length of perpendicular
from origin to the non-vertical line 𝑙
and 𝛼 is the inclination of 𝑝, then the
equation of the line l is given by
𝑥 cos ∝ + 𝑦 sin ∝ = 𝑝
28

29. Angle between two lines
The angle between two intersecting (non-parallel) and non
perpendicular lines 𝑦 = 𝑚1 𝑥 + 𝐶1 and 𝑦 = 𝑚2 𝑥 + 𝐶2 is given
by
tan 𝜃 =
𝑚1 − 𝑚2
1 + 𝑚1 𝑚2
As tan 180 − 𝜃 = − tan 𝜃, when 𝜃 is the acute angle, we can
take the modulus of the value in the above expression.
29

30. Sin, cos and tan values
𝟎° 𝟑𝟎° 𝟒𝟓° 𝟔𝟎° 𝟗𝟎°
Sin 0
1
2
1
2
3
2
1
Cos
1 3
2
1
2
1
2
0
Tan
0 1
3
1
3 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
30

31. Angle between two lines
Notes:
If lines are parallel
tan 𝜃 = 0 ⇒ 𝑚1 = 𝑚2
If lines are perpendicular
tan 𝜃 = tan(
𝜋
2
) = ∝
1 + 𝑚1 𝑚2 = 0 ⇒ 𝑚1 𝑚2 = – 1
31

32. Angle between two lines - Problems
Find the acute angle between two lines 𝐿1 and 𝐿2,
where 𝐿1 is parallel to the x-axis and 𝐿2 has a slope 1
1. 30 degrees
2. 45 degrees
3. 90 degrees
4. None of these
32
Answer : 45 degrees

33. Distance of a point from a line
The length of the perpendicular from a given point
(𝑥1, 𝑦1) on the line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 is given by
𝑎𝑥1 + 𝑏𝑦1 + 𝑐
𝑎2 + 𝑏2
Special case: when the point is the origin
𝑐
𝑎2 + 𝑏2
33

34. Distance between two Parallel lines
The distance between two parallel lines 𝐿1 and 𝐿2 is given
by
𝐿1 → 𝑎𝑥 + 𝑏𝑦 + 𝑐1 = 0
𝐿1 → 𝑎𝑥 + 𝑏𝑦 + 𝑐2 = 0
𝑐1 − 𝑐2
𝑎2 + 𝑏2
34

35. Distance between two Parallel lines - Problem
A straight line through the origin O meets the parallel
lines 4𝑥 + 2𝑦 = 9 and 2𝑥 + 𝑦 + 6 = 0 at points P and Q
respectively. Then, the point O divides the segment PQ in
the ratio
(a) 1:2
(b) 3:4
(c) 2:1
(d) 4:3
Answer : 3:4
35

36. Conditions of Collinearity of three Points
1. Area of triangle ABC is zero.
2. Slope of AB = Slope of BC = Slope of AC
3. Distance between A and B + Distance between B and C =
Distance between A and C
36

37. Finding Circumcenter of a Triangle
1. If ‘O’ is the circumcentre of a triangle ABC, then
OA=OB=OC and OA is called the circumradius.
2. To find the circumcentre of ∆𝐴𝐵𝐶, we use the relation
OA=OB=OC. Form two linear equations and solve them
to find the values of the x and y coordinates
37

38. Finding Circumcenter of a Triangle - Problem
Find the coordinates of the circumcenter of a triangle
whose vertices are at 𝐴 3,1 , 𝐵 −1,3 and 𝐶(−3, −3)
1.
2
7
,
4
7
2. −
2
7
, −
4
7
3.
7
2
,
5
2
4. −
7
2
, −
5
2
Answer : option
2
38

39. Finding orthocenter of a Triangle
1. To determine the Orthocentre, first we find equations
of lines passing through vertices and perpendicular to
the opposite sides.
2. Solving any two of these three equations we get the
coordinates.
39

40. Practice Problems
If 𝐴(1,2), 𝐵(2,5), 𝐶(5,7) and 𝐷(4,4) are the four vertices of
a quadrilateral, then the quadrilateral is a
1. Square
2. Parallelogram
3. Rhombus
4. rectangle
Answer : option
2
40

41. Practice Problems
If 𝐴(1,2), 𝐵(4,3), 𝐶(6,6) and 𝐷(𝑥, 𝑦) are the four vertices of
a parallelogram taken in an order, then the value of 𝑥 + 𝑦
is
1. 5
2. 7
3. 8
4. None of these
Answer : option
3
41