tSAINT JOSEPH ACADEMY

OF SAN JOSE, BATANGAS INCORPORATED

JUNIOR HIGH SCHOOL DEPARTMENT

MODULE NUMBER: 6 and 7 FOURTH GRADING

Date: February 28 – March 1 - 4, 2022 S.Y. 2021 – 2022

CLASS NAME:
GRADE and SECTION:
NUMBER: _________________________________
ARIES - GEMINI
________ ___________
SUBJECT:
MATHEMA TEACHER: Mr. Jeffrey D. Pintor CONTACT NUMBERS:
09499128089
TICS 10
SJA Vision Statement SJA Mission Statement

The SJA Administrators, faculty and staff join The SJA, a recognized institution of learning
hands with the parents, alumni and its allies in commits itself for the upliftment, development, and
creating an educational environment that will develop integral growth of its learners. SJA provides learners a
in its learners the 21st century skills necessary to well-rounded education that will maximize their 21st
improve literacy, scientific and technical potentials that century skills and develop their total personality to
embodies love, loyalty and hope for the family, school, prepare them for higher educational pursuits and global
community and country.
competitiveness.
SJA Philosophy Statement

Saint Joseph Academy is a highly respected non-sectarian secondary institution dedicated to impart to the
students the respect in the individual needs of themselves and others. Thus, SJA believes that every student has
the right to learn and get a quality education.

SJA Goals and Objectives

Accepting its role as the second home of its students, SJA endeavors to:

mold its students to be God-loving and God-fearing, in imitation of the virtues of St. Joseph while respecting all
religious beliefs existing in the community.
direct the minds of students to become productive citizen with positive Filipino values, developing in them love
of family, community and country.
strengthen the school-community relations through extension programs
stimulate in each student a desire to maximize his own talent

SJA Core Values

S – Simplicity and Self Discipline (Kasimplehan at Disiplinang Pansarili)

J – Justice (Hustisya)
A – Acceptance and Asssertiveness (Pagtanggap at Pagtitiwala)
E – Excellence and Enthusiasm (Kahusayan at Kasipagan)
R – Rapport and Respect (Pagkakaisa at Paggalang)

- - - - - A STUDENT’S PRAYER - - - - -
Lord Jesus, I dedicate myself to you as a student
Thank you for all your blessings and graces, thank you for my parents, teachers, classmates and my school.
Enlighten me to realize the importance of education.
Always be there to guide me to overcome my faults, failures and frustrations that I may become more pleasing to you.
Cast out all evil spirits from me and all my educational materials and other elements that I may encounter during my
student life.
Help me to learn the right values and be able to achieve my goals in life.
Mold me in my growing years to develop my god –given skills and talents.
Empower me with the “gifts of the holy spirit” especially the gift of wisdom, knowledge and love.
I ask these in the mighty name of Jesus through the powerful intercession of Mama Mary.
MODULE NUMBER: ___2teach me forTHIRD
Yes, Lord Jesus, QUARTER
you are the greatest teacher.
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 tSAINT JOSEPH ACADEMY
OF SAN JOSE, BATANGAS INCORPORATED
JUNIOR HIGH SCHOOL DEPARTMENT

LEARNING TARGETS:

1. Apply the properties of some geometric figures to place them on the coordinate plane:
2. Verify the properties of geometric figures using coordinate geometry: and
3. Solve problems involving geometric figures on the coordinate plane.

CHAPTER
CHAPTER
34 COORDINATE GEOMETRY

Coordinate geometry or analytic geometry is the study of geometric figures by using algebra. The first
systematic presentation of coordinate geometry was given by the French mathematician Rene Descartes
(1637) in La Geometrie. The logical or natural association between algebra and geometry is one of the
great advances in the field of mathematics for it enables mathematicians to give simple proofs for many
difficult problems. In addition, it makes possible the discovery of many geometric fact.

MODULE 6

Week 1

Placing Figures in the Coordinate Plane

When sketching a figure in the coordinate plane where some of the coordinates of the vertices are
expressed as variables, it is generally a good practice to place a vertex at the origin and one side on an
axis. For example, if a proof involves a right triangle, we can rely on the definition of a right triangle in
placing it.
C y C(0,b) y B(c, d)

C(e, f)

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JUNIOR HIGH SCHOOL DEPARTMENT

A B A(0.0) B(a,0) x A(a, b)

Analytic geometry is also known as coordinate geometry. It uses algebraic methods to study geometric
figures on the coordinate plane.
A good choice in placing right triangle ABC in the coordinate plane is shown in 2nd figure. If the triangle were
placed as shown in 3rd figure, the coordinates of the vertices would become much more complicated.
Guidelines for Placing Figures on a Coordinate Plane
1. Use the origin as vertex or center.
2. Place at least one side of a polygon on an axis.
3. Keep the figure in the first quadrant, if possible.
4. Use coordinates that will make computation as simple as possible.

Example 1:
Give the coordinates of points M and P without using any new variable.
a. square b. rhombus c. isosceles trapezoid
M N(b, c)
N(0, a) M N(0, h)
M O(k, 0)

O P P P(-a, 0) O(a, 0)
Solution:
a. Quadrilateral MNOP is a square, hence MN = NO = OP = MP. The length of NO is a units so the lengths
of segments OP, MP and MN also must be a units. Hence, the coordinates of the points are: P(a, 0), M(a, a)
and O(0, 0).
b. Quadrilateral MNOP is a rhombus. Since diagonals of a rhombus bisect each other, the distance of P
from the origin is h and the distance of M from the origin is k. hence, the coordinates of the points are: P(0, -
h) and M(-k, 0).
c. Quadrilateral MNOP is an isosceles trapezoid with segment MN is perpendicular with segment PO.
Since the y-axis is the perpendicular bisector of segment PO, it is also the perpendicular bisector of
segment MN. Hence, the coordinates of point M are (-b, c).
Example 2:
Find the missing coordinates of the following diagrams.
a. Rectangle RECO b. Rectangle ABCD is centered at the origin with sides
R(?, ?) E(a, b) parallel to the axes.

A(?, ?) B(a, b)
O (?, ?) C(?, ?)

D(?, ?) C(?, ?)
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Solution:
a. E has coordinates (a, b). Thus, the coordinates of the other vertices are O(0, 0), R(0, b), and C(a, 0).
b. The vertex has coordinates (a, b). Hence, the coordinates of the other points are A(-a, b), C(a, -b) and
D(-a, -b).
Example 3:
Use the figure below with the indicated coordinates to answer the following:
A(0, m) B(m, m)

O(0, 0) C(m, 0)
a. What kind of quadrilateral is ABCO? Explain.
b. Find the midpoints of segment AC and BO. What do they prove?
c. Find the slope of segment AC and BO. What conclusion can you derive about segment AC and BO?

Solution:
a. Rectangle ABCO is a square. AB = BC = CO = OC = m and angle AOC is a right angle.
0+m m m+ 0 m
b. Midpoint of segment AC: x = = and y = =
2 2 2 2
m+ 0 m m+ 0 m
Midpoint of segment BO: x = = and y = =
2 2 2 2
Both midpoints are (m/2, m/2). This proves that segment AC and BO bisect each other.
0−m
c. Slope of segment AC: m = =1
m−0
0−m
Slope of segment BO: m = =1
0−m
Segment AC and BO are perpendicular.

MODULE 7

Week 2
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 tSAINT JOSEPH ACADEMY
OF SAN JOSE, BATANGAS INCORPORATED
JUNIOR HIGH SCHOOL DEPARTMENT

In this lesson, you will be able to:

1. verify the properties of geometric figures using coordinate geometry; and
2. solve problems involving geometric figures on the coordinate plane.
Exploration:

Coordinates
1. Graph the parallelogram with vertices A (3,in
2), Proof
B (11, 2), C (9, -2), and D (1, -2).
2. Connect the midpoints of the consecutive sides to form quadrilateral PQRS with P the midpoint of
segment AB, Q the midpoint of segment BC, R is the midpoint of segment CD and S is the midpoint of
segment AD.
3. Find:
a. the slope of segment PQ: m =
b. the slope of segment RS: m =
c. the slope of segment PS: m =
d. the slope of segment QR: m =
e. What can you conclude about quadrilateral PQRS?
4. Based on your answer in item number 3e, show that:
a. PQ = RS
PQ =
RS =
b. PS = QR
PS =
QR =
c. Segment PR and Qs bisect each other.
In exploration, you used coordinate geometry to show that the quadrilateral formed by connecting the
midpoints of a parallelogram is also a parallelogram. Coordinate geometry can be used to simplify proofs of
many geometric theorems.
Example 1:
Use coordinate geometry to prove that the midpoint of the hypotenuse of a right triangle is equidistant from
the vertices.
Solution:
Given: Right triangle ABC with Q the midpoint of hypotenuse segment AB
Prove: QA = QB = QC
Proof:
1. Draw right triangle ABC on a coordinate plane. Locate the right angle, angle C, at the origin and leg
segment CA on the x-axis. Conveniently label the coordinates using multiples of 2 because we are dealing
with midpoint.
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 tSAINT JOSEPH ACADEMY
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JUNIOR HIGH SCHOOL DEPARTMENT

B(0, 2b)

Q(a, b)

C (0, 0) A(2a, 0)
2. Using the Midpoint Formula, the coordinates of Q are:

2 a+0
X= = 2a/2 = a
2
0+2 b
Y= = 2b/2 = b
2
3. Show that QA = QB = QC using the Distance Formula:

√
QA = ( 2 a−a )2+(0−b)2 = √ a2 +b 2

QB = √ ( 2 a−a ) +(0−b) = √ a +b
2 2 2 2

QC = √ ( 2 a−a ) +(0−b) = √ a +b
2 2 2 2

Because QA = QB = QC, then the midpoint of the hypotenuse of a right triangle is equidistant from the
vertices of the triangle.
Example 2:
Prove: a. the diagonals of a parallelogram bisect each other.
b. the diagonals of an isosceles trapezoid are congruent.
c. the midsegment of a trapezoid is parallel to the bases.
d. the length of the midsegment of a trapezoid is half the sum of the lengths of the bases.
Solution:
a. Given: quadrilateral OABC is a parallelogram.
Prove: segment OB bisects segment AC, and segment AC bisects segment OB

 Place the parallelogram in the coordinate plane with one vertex at the origin and one side along the
x-axis. Use multiples of 2 to specify coordinates because midpoints are involved.

C(2b, 2c) B(2a + 2b, 2c)

O A(2a, 0)

 Show that CD = AD and OD = Bd.

Find the midpoint of segment AC or BO
2 a+2 b
Using segment x = =a+b
2
y = 2c/2 = c
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JUNIOR HIGH SCHOOL DEPARTMENT

2 a+2 b
Using segment BO: x = =a+b
2
y = 2c/2 = c
The coordinates of the midpoint are (a + b, c).
 Show that OD = BD:
√
OD = ( a+b−2 b )2 +(c−2 c)2 = √ ( a−b)2 +c 2

√
BD = ( a+b−2 b )2 +(c−2 c)2 = √( a−b)2 +c 2

Thus, OD = BD.
Since CD = AD and OD = BD, segment OB bisects segment AC, and segment AC bisects segment
OB.

b. Given: quadrilateral ABCD is a trapezoid with segment AB ≅ segment DC.

Prove: segment BD ≅ segment AC

 Place the isosceles triangle in the coordinate plane as shown on figure below.

(-b, c)C B(b, c)

(-a, 0)D O A(a, 0)

Show that AC = BD.
Using the Distance Formula:

√
AC = [a−(−b )] 2+(0−c)2 = √ ( a+b)2+ c 2

BD = √ [a−(−b )] +(0−c) = √ ( a+b) + c

2 2 2 2

Since AC = BD, the diagonals of an isosceles trapezoid are congruent.

c. Given: segment DE is the midsegment of trapezoid OABC.
Prove: segment DE is parallel with segment BC, segment DE is parallel with segment OA, and DE =
½(AO + BC).
Place the trapezoid in the coordinate plane as shown below.

C(2b, 2c) B(2d, 2c)

D E

O A(2a, 0)
Find the coordinates of D and E, the midpoint of segment CO and segment AB, respectively.
Coordinates of D: Coordinates of E:
2b +0 2 a+2 d
x= =b y= =a+d
2 2
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JUNIOR HIGH SCHOOL DEPARTMENT

2 c+0 0+2 c
x= =c y= =c
2 2
Thus, we have D(b, c) and E(a + d, c).
Find the slopes of segments BC, AO, and DE.
Slope of segment BC: Slope of segment DE
2 c−2c c −c
m= =0 m= =0
2 d−2 b a+d −b

Slope of segment AO:

0−0
m= =0
2 a−0
Thus, segment DE is parallel with segment BC is parallel with segment AO.
d. Find BC, A), and DE using the distance formula.

√
BC = ( 2 d−2 b ) ¿2 +(2 c−2 c )2 ¿ = √(2 d−2b)2 = 2d – 2b

AO = √ ( 2 a−0 ) ¿ −(0−0) ¿ = √ (2 a) = 2a
2 2 2

DE = √ ( a+ d−b ) ¿ +(c−c ) ¿ = √ (a+d−b)

2 2 2
=a+d–b

Show that DE = ½(AO + BC)

a+ d – b = ½(2a + 2d – 2b)
a+d–b=a+d–b
Thus, DE = ½(AO + BC).

WEEK 1

Activity 1:
Find the missing coordinates for each figure without using any new variables.

1. square 2. square
(0, m) A
A (n, n)

B (n, - n)
B
3. rectangle 4. Rhombus A

A (m, n) (-m, 0) B
-

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JUNIOR HIGH SCHOOL DEPARTMENT

O B (0, -n)

5. isosceles trapezoid
A B(n, q)

(-m, 0) (m, 0)

Activity 2:
1.a. Draw a square PEAT which diagonals are of length 2k and lie on the axes.
b. Find the coordinates of P, E, A, and T.
c. Find PE.
d. Find the slope of segments PE and EA.
e. Compare the slopes in d.
2.a. Draw an isosceles triangle with base length 2m and height 2n. Place the base on the s-axis and one of
the vertices in the origin.
b. Find the lengths of the sides of the triangle.
3.a. Draw another isosceles triangle with base length and height equal to the isosceles triangle in item
number 2. Place the base on the x-axis and its midpoint at the origin.
b. Find the lengths of the legs of the triangle.
c. Compare the lengths of the sides of the isosceles triangle with the sides in item number 2.

WEEK 2:

Activity 1:
Perform as indicated.
1. Show that the diagonals of rhombus SAIL are perpendicular to each other.
A(25,40) I(65, 35)

S(5, 5) L(45, 5)

2. Show that one pair of sides of trapezoid OILB are parallel lines.
L(40, 30)

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B(49, 25)

I(10,15)
O915, 8)

3. Show one diagonal of a kite is the perpendicular bisector of the other diagonal.
T(8, 4)
I(5, 5)

E(7, 1)
K(0, 0)

Activity 2:
Prove the following using coordinate geometry:
1. The diagonals of a rectangle are congruent.
2. If the diagonals of a parallelogram are congruent, them it is a rectangle.
3. The midsegment of an isosceles triangle form another isosceles triangle.
4. The segments joining the midpoints of the sides of a rectangle form a rhombus,

Reference: E – MATH GRADE 1O

By: ORLANDO A. ORONCE AND MARILYN O. MENDOZA

P a g e 10