This document discusses the distance formula and how to find the distance between two points on a coordinate plane. It begins by defining key concepts like horizontal and vertical distance. Examples are provided to demonstrate using the distance formula to calculate distances. The document then shows an example application problem involving finding the distance between two cities on a map. It concludes by generalizing the process for finding distances between any two points whether aligned horizontally/vertically or not.

1. MATHEMATICS
10
THE DISTANCE FORMULA
Subject Matter
LEIDEL CLAUDE Z. TOLENTINO
Subject Teacher

2. LEARNING
OBJECTIVES
a. derive the distance formula.
b. Find the distance between a pair
of points on the coordinate plane.
c. Appreciate how the distance
formula facilitates finding solutions
to real-life problems.
2

3. Directions: Give the difference in each
pair of integers.
1. -7 – (-4)
3
ACTIVITY 1 AS QUICK AS YOU CAN!
3. -24 – 11
4. 9 – (-19)
5. 21 – 5
2. 12 – (- 14)
-3
26
-35
28
16

4. 20XX Pitch Deck 4
Along Aguinaldo Highway
are the houses of four Grade
10 students namely Jose,
Emilio, Gabriella and
Antonio. Their teacher’s
house is some blocks away
from Gabriella’s house as
illustrated:
Activity 2

5. 20XX Pitch Deck 5
1. What is the distance
between Emilio’s
house and
Gabriella’s house?
3 units
4 units
2. What is the distance
between their teacher’s
house and Gabriella’s
house?

6. 20XX Pitch Deck 6
3 units
4 units
How did you find the
distance of Emilio’s house
and Gabriella’s
house?Distance of
Teacher’s house and
Gabriella’s house?

7. 20XX Pitch Deck 7
1. Give the coordinates of
points Emilio’s house,
teacher’s house, and
Gabriella’s house.

8. 20XX Pitch Deck 8

9. 20XX Pitch Deck 9
(4, 5)
(4, 1)
(1, 1)

10. 20XX Pitch Deck 10
2. Plot the coordinates representing
the houses of Emilio (E), their
teacher (T), and Gabriella (G).

11. 20XX Pitch Deck 11
T(4, 5)
E(1, 1) G(4, 1)

12. 20XX Pitch Deck 12
3. Use line segments to connect
the points of Emilio (E), their
teacher (T), and Gabriella (G).
What kind of triangle is formed?

13. 20XX Pitch Deck 13
T(4, 5)
E(1, 1) G(4, 1)

14. 20XX Pitch Deck 14
4. Using the coordinates, how do
you determine the horizontal
distance from E to G or G to E? How
do you determine the vertical
distance from T to G or G to T?

15. 20XX Pitch Deck 15
T(4, 5)
E(1, 1) G(4, 1)
EG= ∣1-
4∣=3
EG= ∣4-
1∣=3
TG= ∣5-1∣=4
GT= ∣1-5∣=4

16. 20XX Pitch Deck 16
5. How to find the distance between
the teacher’s house(T) and Emilio’s
house(E)?
The distance of the house of Emilio,
and the teacher from each other can
be determined by applying the
Pythagorean Theorem 𝑐2= 𝑎2+ 𝑏2

17. 20XX Pitch Deck 17
𝑐2= 𝑎2+ 𝑏2
𝑆𝑂𝐿𝑈𝑇𝐼𝑂𝑁:
𝑐2= 32+ 42
𝑐2= 9 + 16
𝑐2
=
25
𝑐2= 25
C = 5

18. 20XX Pitch Deck 18
6.Replace the coordinates of E by (x1, y1)
and T by (x2, y2).
What would be the resulting
coordinates of G?

19. 20XX Pitch Deck 19
T(x2, y2)
E(x1, y1) G(x2, y1 )

20. 20XX Pitch Deck 20
7. What expression represents
the distance between E and
G?

21. 20XX Pitch Deck 21
T(x2, y2)
E(x1, y1) G(x2, y1 )
EG= ∣ x1 - x2
∣
GE= ∣ x2 – x1
∣

22. 20XX Pitch Deck 22
8. How about the expression that
represents the distance between T and
G?

23. 20XX Pitch Deck 23
T(x2, y2)
E(x1, y1) G(x2, y1 )
TG= ∣ y2 – y1
∣
GT= ∣ y1 - y2
∣

24. 9.What equation will you use to find the
distance between E and T? Explain your
answer.
𝑐2= 𝑎2+ 𝑏2 𝐸𝑇2= 𝐸𝐺2+ 𝑇𝐺2

25. 20XX Pitch Deck 25
𝐸𝑇2
= (x2 – x1)
2
+ (y2 – y1)
2 TG= ∣ y2 – y1
∣
GE= ∣ x2 – x1
∣
𝐸𝑇2= 𝐺𝐸2+ 𝑇𝐺2
𝐸𝑇2= (x2 – x1)
2
+ (y2 – y1)
2
ET = (x2 – x1)
2
+ (y2 – y1)
2
d = (x2 – x1)
2
+ (y2 – y1)
2

26. DISTANCE BETWEEN TWO
POINTS
26
The distance between two points is
always nonnegative.
The horizontal distance between these points is the
absolute value of the difference of their x-coordinates.
Likewise, the vertical distance between these points is
the absolute value of the difference of their y-
coordinates.
The distance between two points, whether or not they
are aligned horizontally or vertically, can be determined
using the distance formula.

27. GROUP
WORK
Find the distance between the two given
points
1.P (1, 3) and Q(7, 11) GROUP 1
2.E (1, 1) and T (4, 5) GROUP 2
3. A(-2, 1) and B(3, 3) GROUP 3
20XX Pitch Deck 27

28. DISTANCE BETWEEN TWO
POINTS
1. Find the distance between P (1, 3) and Q(7, 11)
PQ
=
(𝑥2−𝑥1)2 + (𝑦2−𝑦1)2
(7 − 1)2 + (11 − 3)2
=
(6)2 + (8)2
=
36 + 64
=
100
=
10 𝑢𝑛𝑖𝑡𝑠
PQ
=
𝑥1 𝑥21
𝑦1 𝑦2

29. DISTANCE BETWEEN TWO
POINTS
2. Find the distance between P (1, 1) and Q(4, 5)
ET
=
(𝑥2−𝑥1)2 + (𝑦2−𝑦1)2
(4 − 1)2 + (5 − 1)2
=
(3)2 + (4)2
=
9 + 16
=
25
=
5 𝑢𝑛𝑖𝑡𝑠
ET
=
𝑥1 𝑥21
𝑦1 𝑦2

30. DISTANCE BETWEEN TWO
POINTS
3. Find the distance between A (-2, 1) and B(3, 3)
AB
=
(𝑥2−𝑥1)2 + (𝑦2−𝑦1)2
[3 − (−2)]2 + (3 − 1)2
=
(5)2 + (2)2
=
25 + 4
=
29 or5.38 units
AB
=
𝑥1 𝑥21
𝑦1 𝑦2

31. 31
APPLICATION:
A map showing the locations of different municipalities and cities
is drawn on a coordinate plane. Each unit on the coordinate
plane is equivalent to 6 kilometers. Suppose the coordinates of
Mabini City is (2, 2) and Sta. Lucia town is (6, 8). What is the
shortest distance between these two places?

32. 32
APPLICATION: d= (𝑥2−𝑥1)2 + (𝑦2−𝑦1)2
= (6 − 2)2 + (8 − 2)2
= (4)2 + (6)2
= 16 + 36
= 16 + 36
= 52
= 52

33. 33
APPLICATION:
= 52
= (4)(13)
= (4)(13)
d =2 (13) units
Since 1 unit in the coordinate
plane is 6 units, multiply the
obtained value of distance by 6 to
get the distance between Sta.
Lucia to Mabini City
= (2 13)(6)
12 13 or 43.67units

34. PLUS-MINUS-INTERESTING
CHART
PLUS
Students place all
positive ideas
-good points
MINUS
Negative ideas
-bad points
INTERESTIN
G
-interesting ideas you
learned
-what you find
interesting about the
lesson
20XX Pitch Deck 34

35. PLUS-MINUS-INTERESTING
CHART
PLUS MINUS
INTERESTIN
G
What is the importance in knowing the distance?

36. 36
GENERALIZATION:
How can we find the horizontal
distance, vertical distance, and distance
of two points whether or not they are
aligned horizontally and vertically?

37. DISTANCE BETWEEN TWO
POINTS
37
The distance between two points is
always nonnegative.
The horizontal distance between these points is the
absolute value of the difference of their x-coordinates.
Likewise, the vertical distance between these points is
the absolute value of the difference of their y-
coordinates.
The distance between two points, whether or not they
are aligned horizontally or vertically, can be determined
using the distance formula.

38. 38
EVALUATION:
Find the distance between each pair of points
on the coordinate plane. (5 points each)
1. X(2, -3) and Y(10, -3)
2. A(3, -7) and B(3, 8)

39. 39
EVALUATION: Answer
1. XY= (10 − 2)2 + [−3 − (−3)]2
= (8)2 + (0)2
= 64 + (0)
= 64
= 8 units
2. AB= (3 − 3)2 + (8 − (−7)2
= (3 − 3)2 + (8 − (−7)2
= (0)2 + (15)2
= 225
= 15 units

40. 40
ASSIGNMENT:
1. What is midpoint? midpoint formula?

41. -11 11
0 1 2 3 6
4 5 9
7 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10
A B C D E F G
Activity 2 How Long Is This Part?
Directions: Use the number line below to find
the length of each of the following segments.
1. 𝐴𝐵
2. 𝐵𝐶
3. 𝐶𝐷
4. 𝐷𝐸
5. 𝐸𝐹
6. 𝐹𝐺
4 units
4 units
6 units
2 units
3 units
1 unit

42. 1. How did you find the length of
each segment?
-11 11
0 1 2 3 6
4 5 9
7 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10
A B C D E F G
Ans: Counting the number of
units from one point to the other

43. 2. Did you use the coordinates of
the points in finding the length of
each segment? If yes, how?
-11 11
0 1 2 3 6
4 5 9
7 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10
A B C D E F G
Ans: yes, by finding the absolute
value of the DIFFERENCE of the
coordinates of the points.

44. 3. Which segments are
congruent? Why?
-11 11
0 1 2 3 6
4 5 9
7 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10
A B C D E F G
Ans:
𝐴𝐵 ≅ 𝐵𝐶 , 𝐴𝐶 ≅ 𝐶𝐸 , 𝐶𝐷 ≅ 𝐷𝐺 , 𝐴𝐵 ≅ 𝐸𝐺

45. 4. How would you relate the
lengths of the following segments?
a) AB, BC , and AC
b) AC , CE , and AE
-11 11
0 1 2 3 6
4 5 9
7 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10
A B C D E F G
AB + BC = AC
AC + CE = AE

46. 5. Is the length of AD the same as the
length of DA? How about BF and FB ?
Explain your answer.
-11 11
0 1 2 3 6
4 5 9
7 8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10
A B C D E F G
Ans:
𝐴𝐷 =∣ −10 − 4 ∣ = 14 𝐷𝐴= ∣ 4 − (−10) ∣ =14

47. ACTIVITY 3 LET ME FORMULATE
DIRECTIONS: PERFORM THE FOLLOWING ACTIVITY.
AND ANSWER THE QUESTIONS THAT FOLLOWS.
1.Plot the points A(2,1) and B(8,9) on the
coordinate plane below.
2. Draw a horizontal line passing through
A and a vertical line containing B.

48. ACTIVITY 4 LET ME FORMULATE
DIRECTIONS: PERFORM THE FOLLOWING ACTIVITY.
AND ANSWER THE QUESTIONS THAT FOLLOWS.
3. Mark and label the point of intersection of the
two lines as C.
What are the coordinates of C? Explain how
you obtained your answer.
What is the distance between A and C?
How about the distance between B and C?

49. ACTIVITY 4 LET ME FORMULATE
DIRECTIONS: PERFORM THE FOLLOWING ACTIVITY.
AND ANSWER THE QUESTIONS THAT FOLLOWS.
4. Connect A and B by a line segment. What
kind of triangle is formed by A, B, and C?
Explain your answer.
How will you find the distance between A
and B?
What is AB equal to?

50. ACTIVITY 4 LET ME FORMULATE
DIRECTIONS: PERFORM THE FOLLOWING ACTIVITY.
AND ANSWER THE QUESTIONS THAT FOLLOWS.
5. Replace the coordinates of A by (𝑥1, 𝑦1) and B by (𝑥2,𝑦2).
What would be the resulting coordinates of C?
What expression represents the distance between A and C?
How about the expression that represents the distance
between B and C?
What equation will you use to find the distance between A
and B? Explain your answer.