Math 10th grade
LEARNING UNIT
Let us discover new ways
and use the Cartesian
Plane.
S/K
LEARNING OBJECT
Description of the circumference
1. SCO: Identify the use of the circumference on
everyday tasks.
 SKILL 1.1 Recognize the circumference used in
mechanical components, arts, design, etc.
 SKILL 1.2 Recognize the reflective property of
the circumference.
2 SCO: Geometrically represent the circumference.
 SKILL 2.1 Draw circumferences by establishing
some homemade strategies.
 SKILL 2.2 Identify which elements are necessary
in order to draw a circle with specific
characteristics.
 SKILL 2.3 Describe and make use of a compass
to draw circles.
 SKILL 2.4 Draw circumferences on the Cartesian
plane with specific center and radius.
3 SCO: Describe the circumference as a locus.
 SKILL 3.1 Identify the basic elements of a circle,
such as center and radius.
 SKILL 3.2 Identify the relationship between the
distance from a point of the circumference to the
center.
 SKILL 3.3 Characterize each of the points that
make up a circumference
 SKILL 3.4 Create the definition of a
circumference
 SKILL 3.5 Build a circumference on the Cartesian
plane from the radius and the center.
4 SCO: Represent a circumference through an
equation.
 SKILL 4.1 Represent circumferences on the
Cartesian Plane.
 SKILL 4.2 Recognize the process of constructing
the general equation of the circumference,
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Language
Socio cultural context of
the LO
Curricular axis
Standard competencies
Background Knowledge
Basic Learning
Rights
English Review topic
Vocabulary box
located on the Cartesian Plane with its center at
the origin.
SKILL 4.3 Recognize the process of construction
of the general equation of the circumference
located on the Cartesian Plane with its center
outside the origin.
SKILL 4.4 Represent a circumference on the
Cartesian plane from its equation.
SKILL
4.5
Determine
equations
of
circumferences from the center and the radius.
SKILL 4.6 Identify the tangent to a
circumference, as well as some of their
properties.
SKILL 4.7 Solve problematic situations through a
geometrical interpretation of the circumferences
involved and creating equations of the same, and
use imperative forms as a tool to obtain
information in class.
English
Objects in the environment with a circumferential
shape.
Use of the circumference in the art, sculpture and
mechanics.
School areas
Spatial thinking and geometric systems.
Identify the characteristics and properties of the
conical figures (ellipses, parables, hyperbolae) and
use their properties in problem solving.
Graphics in the Cartesian plane, basic algebraic
operations.
Know geometric properties that define different types
of conic elements (parables, ellipses and hyperbolae)
on the plane and use them to find the general
equations for this type of curves.
Imperatives
Locus, Tangent, Center
NAME: _________________________________________________
GRADE: ________________________________________________
INTRODUCTION
The objects that surround us
At present, the circumference has many applications from the forms
objects like CDs or sewer lids have, to the wheels of cars, bicycles and
almost any means of land transport.
The wheel has been used since ancient times as a basis for being able to
move an infinite variety of objects. Today one of its most important uses
is in the auto and oil industries, in the form of tires, pulleys and sprockets,
among many others.
How would you draw a circumference?
OBJECTIVES
To justify why the circumference is a locus.
1. To recognize the usefulness of items with circular shape.
2. To represent a circumference accurately in geometric form.
3. To create a conception of circumference by identifying its
characteristics as a locus.
4. To use equations to represent circumferences on the Cartesian
Plane
5. To use imperative forms in a conversation.
ACTIVITY 1
SKILL 1.1 Recognize the circumference used in mechanical components,
arts, design, etc.
SKILL 1.2 Recognize the reflective property of the circumference.
SKILL 2.1 Draw circumferences by establishing some homemade
strategies.
SKILL 2.2 Identify which elements are necessary in order to draw a circle
with specific characteristics.
SKILL 2.3 Describe and make use of a compass to draw circles.
Reflective property of the circumference
The circumference is also known as the circular billiard table. This
property consists of taking a point A on the circumference and emit a
beam that bounces on point B, the beams are projected successively,
forming the image shown below:
Circumference in art
Visit the following link to see the example of an exhibition of artist
Richard Long, whose leading role is the circumference
(s.a) [wwwkunstundfilmde]. (2011, November 20).
Richard Long: Berlin Circle & Land Art (exhibition in Hamburger
Bahnhof)
[Video File]. Retrieved from:
https://www.youtube.com/watch?v=tRTtEAT8u-o
Circumference in mechanics:
As we saw in previous units, the circumference plays an important role in
mechanics because it is the axis of operation and design of pulley,
sprockets and wheels, which in turn are fundamental elements in
mechanics.
How to draw a circumference
Below you will find examples of some homemade strategies to draw a
circumference.
1. Look around you for an object that is round and the size you want; for
example, a CD, a glass or a cap of a jar. You should be aware that the
outer edge is smooth. Place the selected object on a sheet of paper, grab
it with one hand and draw the outline with the other.
2. Another way is to tie a string to the pen you are using. Hold the pen
where you want the center of the circle to be (image 6a). Then stretch
the string to determine the radius of the circle and tie a new pen to the
end and trace the circumference.
3. A simpler way and without the use of many resources is holding the
pen with the thumb and index finger. Place the little finger on the paper
where you want the center of the circle to go, subsequently place the
pen on paper and start to draw the circle.
How do you use the compass?
1. First, you must verify that the tip of the compass and the pencil
lead are at the same height. You can do so by closing the compass.
2. Put the tip of the compass where you want the center of the
circumference to go. Keep the compass straight.
3. Open the compass according to the radius you want the
circumference to have.
4. Finally, rotate the compass so the pencil draws the circle.
As we have seen, we need two fundamental elements to draw a circle: the
center and the measure of its radius.
Learning activity
With the help from a partner do the following task:
1. Using the string method:
 Use cardboard.
 Draw a circumference with a radius of 30 cm.
 Then plot concentric circumferences, whose radius is half of the
previous one.
 Draw circles in different parts and create your own work of art.
Activity 2.
SKILL 3.1: Identify the basic elements of a circle, such as center and
radius.
SKILL 3.2: Identify the relationship between the distance from a point of
the circumference to the center.
SKILL 3.3: Characterize each of the points that make up a
circumference
SKILL 3.4: Create the definition of a circumference
SKILL 3.5: Build a circumference on the Cartesian plane from the radius
and the center.
SKILL 4.1: Represent circumferences on the Cartesian Plane
SKILL 2.4: Draw circumferences on the Cartesian plane with specific
center and radius
Elements of a circumference
A circumference has two fundamental elements:
Center: Is the equidistant point from all points of the circumference.
Radius: the distance from the center to any point on the circumference.
We can also associate the diameter, which is the segment that passes
through the center of the circumference and divides it into two equal
semi-circumferences.
Circumferences on the Cartesian Plane
To graph a circle in the Cartesian Plane, we need to know the coordinates
of its center and the radius. For example:
Example 1. Circumference with center at (0.0) and radius of 5. It will be
located in the origin and shall have a radius equal to 5.
Example 2. The circumference with a radius of 2 can be as seen in images
11a or 11b. Overall we would have endless circumferences with a radius
of 2 because there is no restriction in relation to the center. But if we are
given the center coordinates, we would only have a single circumference.
For example, circumference would be the circumference with a radius of
2, whose center is (2, -2).
Circumference would have as the center point (-1, 1) and radius 2.
Remember that...
If the radius of a circle is 5, this means that the distance from the center
to any point of the same will be of 5 units.
Learning activity:
1. In the blank spaces, point out the parts of the circumference:
2. Using a line, match the graphs on the right to the information on
the left (coordinates of the center point and measuring the radius).
Center in (2,0) r=1
Center in (-2,0) r=2
Center in (0,1) r=1
ACTIVITY 3
SKILL 4.2: Recognize the process of constructing the general equation
of the circumference, located on the Cartesian Plane with its center at
the origin.
SKILL 4.3: Recognize the process of construction of the general
equation of the circumference located on the Cartesian Plane with its
center outside the origin.
SKILL 4.4: Represent a circumference on the Cartesian plane from its
equation.
SKILL 4.5: Determine equations of circumferences from the center and
the radius.
Circumference Equation with center in the origin:
If we need to find the equation of the circumference, with center (0.0)
where the radius is 2:
1. We must take into account the given definition of circumference. To
find all the equidistant points to the center, we have to find the
distance between a point and the center of the circumference.
For this specific example, we need to find the distance between the
point (𝑥, 𝑦) and the center (𝑥0 , 𝑦0 ).
2. The algebraic expression to find the distance between any two
points is: √(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2= distance. In the case of the
circumference, instead of distance, the result of the expression will
indicate the radius.
Equation of the circumference
√(𝑥 − 0)2 + (𝑦 − 0)2 = 𝑟
When we solve and replace, we have:
√𝑥 2 + 𝑦 2 = 2,
If we square both sides of the equality, we have:
𝑥 2 + 𝑦 2 = 22
For the particular case of the circumference equation, with its center in
the origin and radius 2, it will be: 𝑥 2 + 𝑦 2 = 22 or 𝑥 2 + 𝑦 2 = 4
Equation of a circumference with its center outside the origin
The equation of a circumference with its center outside the origin is found
with the same equation for distance, so:
√(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2 = 𝑟 When we raise to the square both sides of the
equation we have:
(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2 = 𝑟 2
Where (𝑥0 , 𝑦0 ) is the coordinate of the center point of the circumference.
For example: in the circumference equation with center at (2, −1) and
𝑟𝑎𝑑𝑖𝑜 = 3, the equation is:
2
(𝑥 − 2)2 + (𝑦 − (−1)) = 32
Solving we have:
(𝑥 − 2)2 + (𝑦 + 1)2 = 9
Circumference on the Cartesian plane from its equation
If we have an equation like: (𝑥 − 2)2 + (𝑦 + 6)2 = 16
We can infer that the radius will be 4, since in the equation the radius is
squared.
We take the expressions within the parentheses and math them to zero
to determine the coordinates of the center:

𝑥−2=0
𝑥 = 2 X coordinate in the center

𝑦+6=0
𝑦 = −6 Y coordinate in the center
Therefore, the circumference has its center at (-6) and radius of 4. Your
graph is:
Learning activity
Answer questions 1, 2 and 3 based on image 16:
1. The radius of the circumference with center C is:
a) 𝑟 = 2𝑏𝑟 = 4𝑐𝑟 = 0.5𝑑𝑟 = 1
Complete the following sentence:
2. The center of the circumference has ( ) as its coordinates, and the
diameter measures ______________
3.
a)
b)
c)
d)
The circumference equation is:
(𝑥 + 1)2 + (𝑦 − 1)2 = 2
(𝑥 + 1)2 + (𝑦 + 1)2 = 1
(𝑥 + 1)2 + (𝑦 − 1)2 = 1
(𝑥 − 1)2 + (𝑦 + 1)2 = 2
4. The graphical representation of the equation: (𝑥 − 1)2 + (𝑦 − 3)2 = 1
is:
ACTIVITY 4
SKILL 4.6: Identify the tangent to a circumference, as well as some of
their properties.
SKILL 4.7: Solve problematic situations through a geometrical
interpretation of the circumferences involved and creating equations of
the same, and use imperative forms as a tool to obtain information in
class
The tangent line to a circumference
A tangent line to a circumference is one that touches the circumference
at a single point. See Image 17:
As seen in image, a tangent line has the property to be perpendicular to
the radius of the circumference.
Remember that...
To determine if a point belongs to the circumference, it is only necessary
to replace the coordinates in the equation and check the equality. For
example, we are going to check the tangent point (100, 150). Replacing
(100 − 100)2 + (150 − 100)2 = 502 we obtain (50)2 = 502
Problem
In a park you want to build a circular shape playground as shown in the
image. The manufacturer submits to the neighborhood board the outline
of how the play area would be distributed using mathematical analysis,
indicating that the design is to scale, where each grid is equivalent to 50
meters.
The constructor takes as its reference point the intersection of two streets
to present data for the playground.
The center of the park will be at coordinates (100, 100). From this point
we must centrally locate the park, for which the radius r = 50 m
Since we know that 𝑟 = 50 and the coordinates of the center point, the
equation of the circumference is:
(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2 = 𝑟 2
(𝑥 − 100)2 + (𝑦 − 100)2 = 502
In addition, the manufacturer indicates that the mesh that covers the
playground should be tangent at some points in order to form the square
requested by the board. The tangent points are (100, 50); (150, 100);
(100, 150); (50, 100).
LEARNING ACTIVITY:
Answer the questions 1, and 2 based on the following graph:
1. The tangent line to the circumference is:
a) Line T b) Line Q
C) Line N d) Line M
2. Segment N is the __________________ of the circumference.
SUMMARY
HOMEWORK
In groups of 3, answer:
1. According to the following image:
a) Find the radius of the circumference.
b) Find the coordinate of the center point of the circumference.
c) Determine the circumference equation.
d) Draw three straight lines tangential to the previous image.
e) Write the coordinates of the points on which the line touches
the circumference.
2. Based on the following equations, determine:
a)(𝑥 + 10)2 + (𝑦 − 3)2 = 25
b)(𝑥 − 3)2 + 𝑦 2 = 12
c)(𝑥 + 1)2 + (𝑦 − 6)2 = 7

What is the radio?

The coordinates of the center point of each.

Graph on the Cartesian Plane.
3. Search for pictures of paintings and sculptures by local or globally
recognized artists, whose works have circumferences. Develops an
artistic exhibition, with the name of the artist to which each work
corresponds to.
4. An architect draws the plane to a football stadium that they want to
build. He has the design to scale, knows that the diameter shall be
400 m, while the center, by comparing it with the Cartesian Plane on
which he drew it, is located at point (3,5).
a) Find the measurement of the radius of the stadium.
b) Find the equation that describes the stadium mentioned.
EVALUATION
Fill the banks with the missing word:
1. The distance between the center of the circumference and any
point of the same is called: ____________________.
2. A _________________ line is one that cuts the circumference
at a single point.
3. The _______________ of the circumference is equal to twice the
_______________ of the same.
4. Regarding the circumference equation: (𝑥 − 4)2 + (𝑦 + 3)2 = 4, It is
not correct to say that:
a)
b)
c)
d)
The
The
The
The
radius is 2.
center is at (4, - 3).
diameter is 4.
center is: (- 4, 3).
Answer true or false:
5. If a circumference has its center point in the origin and diameter = 36
a) The radio will be 18 _____________
b) The equation will be: 𝑥 2 + 𝑦 2 = 182 __________
BIBLIOGRAPHY
Wikihow. (2010). Cómo trazar una circunferencia. Retrieved on March 19,
2016 from: http://es.wikihow.com/dibujar-un-c%C3%ADrculo.
(s.a). (2009). Circunferencia. Retrived on March 20, 2016 from:
http://ebrgeometria.blogspot.com.co/2009/11/circunferencia.html
(s.a) (s.f). Círculo y circunferencia. Retrieved on March 20, 2016 from:
http://mate.ingenieria.usac.edu.gt/archivos/GeometriaPrecalculo02Capitulo_2b.pdf
GLOSSARY
Circumference: locus of all points in the plane that are equidistant to a
point called center.
Diameter: segment or straight line that divides the circumference into two
equal semi-circumferences. The diameter shall be equal to twice the
measure of the radius.
Radius: the distance from the center to any point on the circumference.
VOCABULARY BOX
Locus: The set of all points that share a property
Tangent: A line that just touches a curve at one point, without cutting
across it.
Center: The middle. Such as the center of a circle or square.
ENGLISH REVIEW TOPIC
Imperatives