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
14. 1. What is the distance between Emilioโs
house and Gabriellaโs house? How about the
distance between their teacherโs house and
Gabriellaโs house?
Along Aguinaldo
Highway are the
houses of four
Grade 10
students namely
Jose, Emilio,
Gabriella and
Antonio. Their
teacherโs house
is some blocks
away from
Gabriellaโs 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
teacherโs house
is some blocks
away from
Gabriellaโs 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
teacherโs house
is some blocks
away from
Gabriellaโs 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
teacherโs house
is some blocks
away from
Gabriellaโs house
as illustrated at
the right.
18. 5. Use the Pythagorean Theorem to find the
distance between the teacherโs house and
Emilioโs house.
Along Aguinaldo
Highway are the
houses of four
Grade 10
students namely
Jose, Emilio,
Gabriella and
Antonio. Their
teacherโs house
is some blocks
away from
Gabriellaโs house
as illustrated at
the right.
20. 1. What is the distance between Emilioโs
house and Gabriellaโs house? How about the
distance between their teacherโs house and
Gabriellaโs house?
Along Aguinaldo
Highway are the
houses of four
Grade 10
students namely
Jose, Emilio,
Gabriella and
Antonio. Their
teacherโs house
is some blocks
away from
Gabriellaโs house
as illustrated at
the right.
Emilioโs to Gabriellaโs house
3 units
Teacherโs to Gabriellaโs 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
teacherโs house
is some blocks
away from
Gabriellaโs 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
teacherโs house
is some blocks
away from
Gabriellaโs 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
teacherโs house
is some blocks
away from
Gabriellaโs house
as illustrated at
the right.
๐ โ ๐ ๐ โ ๐
๐ โ ๐ ๐ โ ๐
24. 5. Use the Pythagorean Theorem to find the
distance between the teacherโs house and
Emilioโs house.
Along Aguinaldo
Highway are the
houses of four
Grade 10
students namely
Jose, Emilio,
Gabriella and
Antonio. Their
teacherโs house
is some blocks
away from
Gabriellaโs house
as illustrated at
the right.
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.
34. Replace the coordinates of E by (๐ ๐, ๐ ๐) and T
by (๐ ๐, ๐ ๐) . The coordinates of G would be
(๐ ๐, ๐ ๐)
The
horizontal
distance
from E to G
is the same
as the
distance
from G to
E.
35. Replace the coordinates of E by (๐ ๐, ๐ ๐) and T
by (๐ ๐, ๐ ๐) . The coordinates of G would be
(๐ ๐, ๐ ๐)
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 (๐ ๐, ๐ ๐) and T
by (๐ ๐, ๐ ๐) . The coordinates of G would be
(๐ ๐, ๐ ๐)
Therefore, the
distance
between these
points is the
absolute value
of the
difference of
their x-
coordinates
(horizontal) or
y-coordinates
(vertical).
37. ๐๐ = (๐ ๐ โ ๐ ๐) ๐๐ (๐ ๐ โ ๐ ๐)
๐๐ = (๐ ๐ โ ๐ ๐) ๐๐ (๐ ๐ โ ๐ ๐)
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 (๐ฅ1, ๐ฆ1) = (1, 1) and ๐ฅ2, ๐ฆ2 = (4, 5).
Substitute the corresponding values of ๐ฅ1, ๐ฆ1, ๐ฅ2and
๐ฆ2 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 (๐ฅ1, ๐ฆ1) = (-3, 1) and ๐ฅ2, ๐ฆ2 = (2,-1).
Substitute the corresponding values
of ๐ฅ1, ๐ฆ1, ๐ฅ2and ๐ฆ2 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?
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.
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.
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
๐ units
5 units
๐๐ units
๐๐ units
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(๐ฅ1, ๐ฆ1) and Q (๐ฅ2, ๐ฆ2) are points on a
coordinate plane.
What is the DISTANCE
FORMULA?
PQ
d
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. โ261 units
I. 9 units
N. 4โ5 units
K. 2โ17 units
A. โ51.25 units
T. โ421 units
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.
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. ๐ = โ๐ โ ๐ ๐ + ๐ + ๐ ๐
b. ๐ = โ๐ โ ๐ ๐ + ๐ + ๐ ๐
c. ๐ = โ๐ + ๐ ๐ + ๐ โ ๐ ๐
d. ๐ = ๐ + ๐ ๐ + ๐ + ๐ ๐
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
๐ ๐๐?
a. 4 c. 8
b. 6 d. 10
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.
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. ๐ = โ๐ โ ๐ ๐ + ๐ + ๐ ๐
b. ๐ = โ๐ โ ๐ ๐ + ๐ + ๐ ๐
c. ๐ = โ๐ + ๐ ๐ + ๐ โ ๐ ๐
d. ๐ = ๐ + ๐ ๐ + ๐ + ๐ ๐
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
๐ ๐๐?
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 = โ(9โ1)2 +(1โ9)2
b. RJ = โ(1โ9)2 +(1โ9)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. 7โ2 units
2. 5โ17 units
3. a. RJ = โ(9โ1)2 +(1โ9)2
4. Yes.
AC=โ26 units; BC=โ32 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 2โ5 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?