This document contains a math lesson on deriving and using the distance formula. It begins by showing how to derive the formula for finding the distance between two points on a coordinate plane by replacing the coordinates with variables. Examples are then shown of applying the formula to find distances between points with given coordinates. The document emphasizes that the distance formula can be used to find the distance between any two points on a coordinate plane.

1. MATH 10DAISYREE JEAN R. MEDINO

2. SPEED TEST #1
ANSWER ON YOUR NOTEBOOK!

3. As Quick As You Can!
MENTAL MATH

4. DIRECTIONS: Give the difference in
each pair of integers.
1. -7 and -4
2. 12 and - 14
3. -24 and 11
4. 9 and -19
5. 21 and 5

5. HONESTY CHECK

6. DIRECTIONS: Give the difference in
each pair of integers.
1. -7 and -4
2. 12 and - 14
3. -24 and 11
4. 9 and -19
5. 21 and 5
= -2
= 26
= -35
= 28
= 16

7. WHO GOT THE SCORE OF…

8. WHO GOT THE SCORE OF…

9. WHO GOT THE SCORE OF…

10. WHO GOT THE SCORE OF…

11. WHO GOT THE SCORE OF…

12. WHO GOT THE SCORE OF…

13. ACTIVITY 1
PLEASE GO TO YOUR ASSIGNED GROUPS!

14. 1. What is the distance between Emilio’s
house and Gabriella’s house? How about the
distance between their teacher’s house and
Gabriella’s house?
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 at
the right.

15. 2. Use line segments to connect the points
representing the houses of Emilio (E), their
teacher (T), and Gabriella (G). What kind of
triangle is formed?
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 at
the right.

16. 3. Give the coordinates of points E, T, and G.
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 at
the right.

17. 4. Using the coordinates, how do you
determine the horizontal distance from E to G
or G to E? the vertical distance from T to G or
G to T?
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 at
the right.

18. 5. Use the Pythagorean Theorem to find the
distance between the teacher’s house and
Emilio’s house.
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 at
the right.

19. LET US CHECK!
MOVE YOUR PAPER TO THE LEFT!

20. 1. What is the distance between Emilio’s
house and Gabriella’s house? How about the
distance between their teacher’s house and
Gabriella’s house?
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 at
the right.
Emilio’s to Gabriella’s house
3 units
Teacher’s to Gabriella’s house
4 units

21. 2. Use line segments to connect the points
representing the houses of Emilio (E), their
teacher (T), and Gabriella (G). What kind of
triangle is formed?
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 at
the right.
RIGHT TRIANGLE

22. 3. Give the coordinates of points E, T, and G.
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 at
the right.
E(1,1)
T(4,5)
G(4,1)

23. 4. Using the coordinates, how do you
determine the horizontal distance from E to G
or G to E? the vertical distance from T to G or
G to T?
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 at
the right.
𝟏 − 𝟒 𝟒 − 𝟏
𝟏 − 𝟓 𝟓 − 𝟏

24. 5. Use the Pythagorean Theorem to find the
distance between the teacher’s house and
Emilio’s house.
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 at
the right.

25. Pythagorean Theorem
𝒄 𝟐
= 𝒂 𝟐
+ 𝒃 𝟐
𝟓 − 𝟏
𝟒 − 𝟏
𝑬𝑻 𝟐
= 𝑬𝑮 𝟐
+ 𝑮𝑻 𝟐
𝑬𝑻 𝟐
= (𝟑) 𝟐
+(𝟒) 𝟐
𝑬𝑻 𝟐
= 𝟗 + 𝟏𝟔
𝑬𝑻 𝟐
= 𝟐𝟓
𝟐
(𝑬𝑻) 𝟐 = 𝟐𝟓
𝑬𝑻 = 𝟓 𝒖𝒏𝒊𝒕𝒔

26. WHO GOT THE SCORE OF…

27. WHO GOT THE SCORE OF…

28. WHO GOT THE SCORE OF…

29. WHO GOT THE SCORE OF…

30. WHO GOT THE SCORE OF…

31. WHO GOT THE SCORE OF…

32. Learning Competency 27: Derives
the distance formula.
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.

33. THE DISTANCE FORMULA
DAISYREE JEAN R. MEDINO

34. Replace the coordinates of E by (𝒙 𝟏, 𝒚 𝟏) and T
by (𝒙 𝟐, 𝒚 𝟐) . The coordinates of G would be
(𝒙 𝟐, 𝒚 𝟏)
The
horizontal
distance
from E to G
is the same
as the
distance
from G to
E.

35. Replace the coordinates of E by (𝒙 𝟏, 𝒚 𝟏) and T
by (𝒙 𝟐, 𝒚 𝟐) . The coordinates of G would be
(𝒙 𝟐, 𝒚 𝟏)
Similarly,
the vertical
distance
from T to G
is the same
as the
distance
from G to T.

36. Replace the coordinates of E by (𝒙 𝟏, 𝒚 𝟏) and T
by (𝒙 𝟐, 𝒚 𝟐) . The coordinates of G would be
(𝒙 𝟐, 𝒚 𝟏)
Therefore, the
distance
between these
points is the
absolute value
of the
difference of
their x-
coordinates
(horizontal) or
y-coordinates
(vertical).

37. 𝐄𝐆 = (𝒙 𝟏 − 𝒙 𝟐) 𝒐𝒓 (𝒙 𝟐 − 𝒙 𝟏)
𝐓𝐆 = (𝒚 𝟏 − 𝒚 𝟐) 𝒐𝒓 (𝒚 𝟐 − 𝒚 𝟏)
The distance from E to T is

38. Illustrative Example 1:
Find the distance between the points E (1, 1) and T (4, 5).
Solution: Let (𝑥1, 𝑦1) = (1, 1) and 𝑥2, 𝑦2 = (4, 5).
Substitute the corresponding values of 𝑥1, 𝑦1, 𝑥2and
𝑦2 in the distance formula, then solve.
The distance between the points E and T is 5
units.

39. Illustrative Example 2:
The Philippines is known for more than 7,107 islands. Many
tourists love travelling by boat from island to island, or island
hopping. Find the shortest distance between the two given islands
in the map beside.
Solution:
Let (𝑥1, 𝑦1) = (-3, 1) and 𝑥2, 𝑦2 = (2,-1).
Substitute the corresponding values
of 𝑥1, 𝑦1, 𝑥2and 𝑦2 in the distance
formula, then solve.
The distance between the two islands is 29
units.

40. What would be the coordinates of K?
What expression represents the
horizontal distance between E and G?
the vertical distance between T and
G?
What equation will you use to find the
distance between E and T?
How do you find the distance
between any pair of points on the
coordinate plane?

41. GROUP WORK

42. Supply the missing parts of the solution by following the
task given in the right column.
1. Find the distance between the points R (4, 3)
and T (10, 11).

43. 2. Find the distance between the points A (-1, 2)
and B (3, -4).
Supply the missing parts of the solution by following the
task given in the right column.

44. EXCHANGE PAPERS!
WRITE CORRECTED BY

45. Supply the missing parts of the solution by following the
task given in the right column.
1. Find the distance between the points R (4, 3)
and T (10, 11).

46. 2. Find the distance between the points A (-1, 2)
and B (3, -4).
Supply the missing parts of the solution by following the
task given in the right column.

47. SEATWORK #1
ANSWER ON YOUR NOTEBOOK

48. DIRECTIONS: Determine the
distance between the
following pairs of points.
1. S(5, -1) and T(5,7)
2. R(0,3) and F(1,4)
3. L(2,6) and B(-3,6)
4. A(-3,1) and M(2,2)
5. P(4,-3) and Q(-1,4)
8 units
𝟐 units
5 units
𝟐𝟔 units
𝟕𝟒 units

49. SUM IT UP
BABALA:
DO NOT COPY!

50. The Distance Formula is an
equation that can be used to find
the distance between any pair of
points on the coordinate plane.
What is DISTANCE FORMULA?

51. if P(𝑥1, 𝑦1) and Q (𝑥2, 𝑦2) are points on a
coordinate plane.
What is the DISTANCE
FORMULA?
PQ
d

52. BOARD WORK

53. The map below shows various landmarks in a town. Find
the distance between the given pairs of landmarks by
using the distance formula. Write the corresponding
letters in the box to answer the question.
E. T. Tirona Memorial National High School and Kalayaan
Hospital

54. I. Island Cove and Binakayan Public Market
N. St. Mary Magdalene School and Municipal Hall
K. Aguinaldo Shrine and Binakayan Public Market
A. Save More and Fire Station
T. Water Camp and Island Cove
What town is formerly known as Cavite
el Viejo?

55. W. √261 units
I. 9 units
N. 4√5 units
K. 2√17 units
A. √51.25 units
T. √421 units

56. QUIZ #1

57. 1. Which of the following is NOT a
formula for finding the distance
between two points on the coordinate
plane?
DIRECTIONS: Read the question and
write the LETTER of the correct answer.

58. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
2.

59. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
3. What is the distance
between points D(-10,2) and
E(6,10)?

60. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
4. Which is NOT the correct way to
apply the distance formula in
finding the distance between
points (-1, 3) and (4, -5)?
a. 𝐝 = −𝟏 − 𝟒 𝟐 + 𝟑 + 𝟓 𝟐
b. 𝐝 = −𝟓 − 𝟑 𝟐 + 𝟒 + 𝟏 𝟐
c. 𝐝 = −𝟏 + 𝟓 𝟐 + 𝟑 − 𝟒 𝟐
d. 𝐝 = 𝟒 + 𝟏 𝟐 + 𝟓 + 𝟒 𝟐

61. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
5. What is x if the distance
between (2, 4) and (x, 10) is
𝟐 𝟏𝟑?
a. 4 c. 8
b. 6 d. 10

62. LET US CHECK
Exchange papers!

63. 1. Which of the following is NOT a
formula for finding the distance
between two points on the coordinate
plane?
DIRECTIONS: Read the question and
write the LETTER of the correct answer.

64. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
2.

65. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
3. What is the distance
between points D(-10,2) and
E(6,10)?

66. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
4. Which is NOT the correct way to
apply the distance formula in
finding the distance between
points (-1, 3) and (4, -5)?
a. 𝐝 = −𝟏 − 𝟒 𝟐 + 𝟑 + 𝟓 𝟐
b. 𝐝 = −𝟓 − 𝟑 𝟐 + 𝟒 + 𝟏 𝟐
c. 𝐝 = −𝟏 + 𝟓 𝟐 + 𝟑 − 𝟒 𝟐
d. 𝐝 = 𝟒 + 𝟏 𝟐 + 𝟓 + 𝟒 𝟐

67. DIRECTIONS: Read the question and
write the LETTER of the correct answer.
5. What is x if the distance
between (2, 4) and (x, 10) is
𝟐 𝟏𝟑?
a. 4 c. 8
b. 6 d. 10

68. Use the distance formula to solve the following.
1. What is the distance between the points M
(3,-6) and P (-4, 1)?
2. Find the length of the line segment whose
endpoints are (3,-12) and (-2, 8).
3. Which will give the correct distance between
R (1, 9) and J (9, 1)?
a. RJ = √(9−1)2 +(1−9)2
b. RJ = √(1−9)2 +(1−9)2
4. Nidora wants to go to a computer shop for
her assignment. There are two shops that are
at coordinates A (4, 3) and B (-5,-2). Her sister
Tinidora suggested that she should go to
computer shop A because it is nearer their
house which is at (-1, 2). Is Tinidora correct?
Justify your answer.

69. 1. 7√2 units
2. 5√17 units
3. a. RJ = √(9−1)2 +(1−9)2
4. Yes.
AC=√26 units; BC=√32 units
AC < BC, therefore computer shop A
is nearer than computer
shop B.

70. ASSIGNMENT
1. Follow Up
Solve the following problem.
Strike knows that the distance between two
points is 2√5 units. If one of the points has
coordinates (3, 0) and the x-coordinate of
the second point is -1, what is the y-
coordinate of the second point? There are
two possible answers.
2. Study: The Midpoint Formula
a. What is the Midpoint Formula?
b. How do you determine the coordinates of
the midpoint of a line segment?