Prerequisite materials: Pythagorean theorems Materials
LESSON PLAN
School
: ………………………..
Subject
: Mathematics
Grade/ semester
: VIII/ II
Number of Meeting : 5 Meetings
Year of Lesson
: 2011/ 2012
Standard of Competency
Deter mine elements and par ts of cir cle and also measur ements
Basic Competence
Deter mine length of tangent of tw o cir cles
Indicators
1. Sketch the gr aph of tangent pass a point and find the pr oper ty of tangent of cir cle
2. Deter mining length of tangent of cir cle
3. Under standing position of tw o cir cles
4. Under standing and sketch inter nal and exter nal common tangents of cir cles
5. Deter mining length of inter nal and exter nal common tangents of cir cle
6. Deter mining the minimum r ope r elate tw o cir cles
Learning Objectives
:
After studying, students ar e able to:
1. Sketch the gr aph of tangent pass a point lie on cir cle and find the proper ty of tangent
2. Sketch the gr aph of tangent pass a point exter nal of cir cle and find the pr oper ty of
tangent
3. Deter mine length of tangent of cir cle
4. Under stand position of two cir cles and identify the cr iter ia for each position
5. Solve applied problem about length of tangent and position of two cir cles
6. Under stand and sketch inter nal and exter nal common tangents of cir cles
7. Deter mine length of inter nal and exter nal common tangents of cir cle
8. Solve applied problem about inter nal and external common tangents
9. Deter mine the minimum rope r elate tw o cir cles or mor e
Prerequisite materials
:
Pythagorean theorems
Materials
:
tangent of cir cle
Headline of t opic
1. Tangent of cir cle int er sect t he cir cle at one and only one point on t he cir cle.
2. Pr oper t y of t angent of cir cle:
a. Each t angent per pendicular t o diamet er or r adius of cir cle
b. Pass a point on cir cle, only a t angent can be cr eat ed
c. Pass a point out side of cir cle, can cr eat e t wo t angent s.
3. Lengt h of t angent of cir cle is squar e r oot of differ ence of squar e dist ance of cent er t o a
point out side t he cir cle and squar e of r adius of t hat cir cle
4. Lengt h of int er nal common t angent is squar e r oot of differ ence of dist ance of t wo
cir cles and squar e r oot of sum of r adii.
5. Lengt h of ext er nal common t angent is squar e r oot of differ ence of squar e of dist ance
t wo cent er point of cir cle and squar e of differ ence of t wo r adii of cir cle
Time Allocations
:
Method/ Approach/ Model
Learning resources
11 x 40 minutes
:
demonstr ation, drill and pr actice, constr uctivism, RME,
PBI
:
Learning Activities
First meeting
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
Phase 7
Phase 8
Phase 9
Phase 10
Phase 11
Student prepares themselves to study. Teacher says learning
objectives and ask some realistic problem of circle’s tangent
Realistic problem of tangent of circle are bicycle’s chain,
pulley, bulldozer’s wheel, etc.
Exploration
Student remember definition of tangent
Tangent is line which is intersects circle exact one point
on the circle.
Student translate how to sketch tangent pass a point on a circle
Student follow teacher explanation about sketching tangent pass a
point on the circle
Sketch tangent pass a point on a circle
Elaboration
Student does exercise page 238 latihan 1 no 1 and 2
Student translate how to sketch tangent pass a point outside the
circle
Student follow teacher explanation about sketching tangent pass a
point outside the circle
Student do exercise page 238 latihan 1 no 3 and 4
Confirmation
Conclude the property of tangent of a circle
Property of tangent of circle:
1. Each tangent perpendicular to one diameter or one radius of
circle
2. Pass a point on circle, only a tangent can be created
3. Pass a point outside of circle, can create two tangents.
Conclude today lesson
Student is given homework
Rubric for exercise
Problem
Indicators
1
Have an idea to sketch tangent
Do the idea
Finding property of tangent based on him/her picture
5 minutes
5 minutes
5 minutes
10 minutes
10 minutes
5 minutes
10 minutes
10 minutes
5 minutes
5 minutes
5 minutes
1 2 3 4
2
Have an idea to sketch tangent
Do the idea
Finding property of tangent based on him/her picture
Rubric for problem sheet
Problem
Indicators
1
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
2
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
3
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
4
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
1= achieve the indicator less than 25%
2= achieve indicator between 25%-50%
3=achieve indicator indicators between 50%-75%
4= achieve indicator more than 75%
1 2 3 4
St udent ’s Problem Sheet
NAME
:
…………..
CLASS
:
………….
LENGTH OF TANGENT OF
CIRCLE
Standard Competence
Determine elements and parts of circle and also measurement
Basic Competence
Determine length of tangent of two circles
Indicator
Determine length of tangent of a circle
Direction!
1. Read summary topic first!
2. Answer exercise based on structure problem solving steps that your teacher explains!
Summary of Topic
Tangent of circle is line which is intersects circle on one point at the circle.
Length of tangent of circle is square root of difference of square distance of center to a point
outside the circle and square of radius of that circle
Exercise
1. Length of tangent of a circle is 24 cm, distance between center to a point outside of circle is
26 cm. Determine radius of circle!
Solve using structure problem solving steps
Given
Ask
Solve
2. Look at the picture!
Given radius of a circle, r = 6 cm and length AB = 10 cm.
Determine length of tangent of circle and find another
tangent of circle pass point B!
Solve using structure problem solving steps
Given
Ask
Solve
3. Determine the area of triangle POQ based on the following picture!
25 cm
O
Q
24 cm
P
Solve using structure problem solving steps
Given
Ask
Solve
O
B
A
4. Two woods which are look like a circle is bundled with 144 cm rope. If radii of two roots are same,
determine the radius!
Given
Ask
Solve
Second Meeting
Phase 1
Student discuss their homework at Erlangga book page 61 no 1-5
Phase 2
Student prepares themselves for studying. Student knows learning
objectives of lesson today
Understanding position of two circles
Phase 3
Exploration
Student make figure of some position of two circle based on
Erlangga book page 62
No
1.
Picture
r2
r1
s
2
r1
.
3.
r1
r2
s
r2
s
15 minutes
5 minutes
20 minutes
4.
r1
r2
s
5.
r1
r2
M
N
6.
s
r1
r2
Phase 4
Phase 5
Student pays attention to teacher’s explanation. Teacher show
some realistic condition of position of two circles and guide
student to conclude position and characteristic of each example
that is given.
1. Position of coin Rp 100 and Rp 500 if it is patched
over.
2. Position of minute hand and hours hand of watch
Elaboration
Student give the name of each position and also its characteristic
No
1.
Picture
r2
r1
Position
Intersection
Caracteristic
r1 + r2 > s
Contiguous
outside
r1 + r2 = s
Contiguous
inside
r1 + r2 > s
s
2.
r1
3.
r1
r2
s
r2
s
5 minutes
30 minutes
4.
r1
s
r2
5.
Disjoint
outside
r1 + r2 < s
Concentric
r 1+r 2=0
Disjoint
inside
r1 + r2 > s
r1
r2
M
N
6.
s
r1
r2
Student do the problem sheet
Confirmation
Discuss the problem sheet
Phase 8
Student know their homework to translate how to sketch internal
and external common tangents
Assessment :
problem sheet
10 minutes
5 minutes
Phase 6
Phase 7
5 minutes
Third Meeting
Phase 1
Phase 2
Phase 3
Phase 4
Student review their knowledge about position of two circle
Show in power point
Student prepares themselves for studying. Student knows learning objectives of lesson today
The students are able to:
1. Understanding internal and external common tangents
2. Sk etching internal and external common tangents
5’
Student translate how to sketch internal common tangents
10’
25’
E xploration
Student sketching internal common tangents
No
Steps
1.
Sketch a circle where center is P
and the radius is r1, and then circle
with central point is Q and radius
is r2.
Picture
r1
P
Q
r2
5’
2.
Join center of both circle
Q
P
3.
Sketch arc of circle with center
point P and point Q with same
radius. Both of arc intersect on
point A and B.
A
P
Q
B
4.
Join point A and point B, so that
its intersect line PQ on point C.
A
Q
P
C
B
5.
Sketch a circle with center point is
point C and diameter CP.
A
P
Q
C
B
6.
7.
Make an arc with radius r1+ r2 and
central point is point P at the top
and at the bootom.
Intersection of these arc are H
and I. Joint point P and H and
also point P and I. Intersection of
line PH and circle P is point D.
Intersection of line PI and circle P
is point E .
D
A
P
F
Q
C
G
B
E
Make arc to circle Q with radius
DQ and the central points are D
D
A
F
Q
P
C
B
E
G
8.
Joint point D and G and also
point E and point F.
Line DG and E F are internal
common tangents.
D
F
A
Q
P
C
G
B
E
Phase 5
Phase 6
Student mention some realistic example of external common tangents
Student sketching external common tangent
No
1.
2.
Steps
Sketch circle with center point is
point P and radius r1, then circle
where center is Q and radius r2.
20’
Picture
P
r1
Q
Join both center of that circle.
P
3.
Sketch arc of circle with center
point on point P and point Q and
have same radius. Both arc
intersect on point A and point B.
Q
A
P
Q
B
4.
Join point A and point B, so line
AB intersect line PQ on point C.
A
P
C
Q
B
5.
r2
Sketch a circle where center is C
and radius CP.
A
P
C
Q
B
6.
Sketch a circle with center point is
point P and radius r1-r2. This circle
intersect circle with central point
is point C on point D and point
E.
A
D
C
P
Q
E
B
7.
Join point P and point D so that
intersect circle with cetral point is
point P and radius r1 on point M.
After that join point P and point
E , so that intersect circle with
center point is point P with radius
r1 on point N.
Name point K and point L for
intersection of circle with the
center is point C and point Q.
Join point M and point K, and
then join point N and point L.
Line KM and line LN are external
common tangent of circle with
center points are point P and
point Q.
8.
M
A
D
C
P
Q
E
L
B
N
M
D
P
A
K
C
Q
E
L
B
N
Phase 7
K
E laboration
10’
Students try to sketch internal and external common tangent for another circles
Phase 8
Confirmation
5’
Students conclude the lesson
Performance Assessment
1. Internal common tangents
No
1.
Steps
Sketch a circle where center is P
and the radius is r1, and then circle
with central point is Q and radius
is r2.
Picture
r1
P
Q
0 1 2
r2
2.
Join center of both circle
Q
P
3.
Sketch arc of circle with center
point P and point Q with same
radius. Both of arc intersect on
point A and B.
A
P
Q
B
4.
Join point A and point B, so that
its intersect line PQ on point C.
A
P
Q
C
B
5.
Sketch a circle with center point is
point C and diameter CP.
A
P
Q
C
B
6.
Make an arc with radius r1+ r2 and ```````````````````````````````````````````````````````
central point is point P at the top ```````````````````````````````````````````````
D
and at the bootom.
A F
Intersection of these arc are H
Q
P
C
and I. Joint point P and H and
also point P and I. Intersection of
B
G
line PH and circle P is point D.
E
Intersection of line PI and circle P
is point E .
7.
Make arc to circle Q with radius
DQ and the central points are D
D
A
F
Q
P
C
B
E
G
8.
Joint point D and G and also
point E and point F.
Line DG and E F are internal
common tangents.
D
F
A
Q
P
C
G
B
E
2. External common tangents
No
1.
2.
Steps
Sketch circle with center point is
point P and radius r1, then circle
where center is Q and radius r2.
0 1 2
Picture
P
r1
Q
Join both center of that circle.
P
3.
Sketch arc of circle with center
point on point P and point Q and
have same radius. Both arc
intersect on point A and point B.
Q
A
P
Q
B
4.
Join point A and point B, so line
AB intersect line PQ on point C.
A
P
C
Q
B
5.
Sketch a circle where center is C
and radius CP.
A
C
P
Q
B
6.
r2
Sketch a circle with center point is
point P and radius r1-r2. This circle
A
D
P
C
Q
E
B
intersect circle with central point
is point C on point D and point
E.
7.
8.
Join point P and point D so that
intersect circle with cetral point is
point P and radius r1 on point M.
After that join point P and point
E , so that intersect circle with
center point is point P with radius
r1 on point N.
Name point K and point L for
intersection of circle with the
center is point C and point Q.
Join point M and point K, and
then join point N and point L.
Line KM and line LN are external
common tangent of circle with
center points are point P and
point Q.
M
A
D
K
C
P
Q
E
L
B
N
M
D
P
A
C
K
Q
E
N
B
L
Note:
0= student don’t do the step
1= student do the step but not suitable
2= student do the step correct
Forth meeting
Phase
Activities
1.
Students pr epar e themselves to study
2.
Explor ation
Students mention some object w hich r elated to inter nal and exter nal
common tangents
3.
Students know the lear ning objectives of the lesson and the r ule of
lesson for today
4.
Students r emind how to sketch inter nal and exter nal common tangent
One of the students sketch the tangents on w hite/ blackboar d
5.
Elabor ation
Teacher br ing students to find r ight triangle on inter nal common
tangent and dir ect students to use the concept of Pythagor ean
theor em to deter mine the length of inner common tangents
Time
5’
5’
5’
5’
6.
7.
8.
9.
By using same w ay, students find the length of exter nal common
tangents
Students solve some pr oblem about inter nal and exter nal common
tangents
Teacher help students to solve their pr oblem
Confir mation
Students discuss the most difficulties pr oblem on class discussion
Student conclude lesson today and know their homework
There are 2 outer common tangents if two circles are disjoint outside
L ength of outer common tangent is square root of difference of square of distance two center
point of circle and square of difference of two radii of circle
Assessment of exercise
1. Radii of two circles are 20 cm and 4 cm. determine the distance between two central points
if known length of outer common tangents is 26 cm.
(1 point)
Given
Radii of two circles are 20 cm and 4 cm.
0.5
Length of outer common tangents is 26 cm
0.5
Ask
Distance between two central points
1
Solve
Suppose
Radius of fist circle = r1= 20 cm
Radius of second circle = r2 = 4 cm
Length of outer common tangents = t = 26 cm
Distance between two center point = s
Length of outer common tangent is square root of difference of square of
distance two center point of circle and square of difference of two radii of
circle
Its mean
−( − )
26 =
− ( 20 − 4 )
26 =
− ( 16 )
2
2
26 =
− 16 2
=
2
2
2
= 26 + 16
= 676 + 256
2
= 932
= √932
s = 30.5
1
1
1
1
2
1
1
1
2. Given radii of two circles are 16 cm and 17 cm. Distance of two central points is 24 cm.
Calculate the length of outer common tangents!
(1point)
Given
Radii of two circles are 16 cm and 17 cm.
0.5
Distance of two central point is 24 cm
0.5
Ask
Distance between two central points
1
Solve
Suppose
Radius of fist circle = r1= 16 cm
Radius of second circle = r2 = 17 cm
1
Distance between two center point = s = 24 cm
Length of outer common tangents = t
Length of outer common tangent is square root of difference of square of
distance two center point of circle and square of difference of two radii of
circle
Its mean
=
=
−( − )
24 − ( 16 − 17 )
24 − ( 1 )
√576 − 1
√575
=
=
=
t = 23.9
1
1
1
1
1
1
3. Suppose radii of two circles are 12 cm and 5 cm. Distance of two central points is 20 cm.
Calculate the length of outer common tangents!
(1 point)
Given
Ask
Solve
Radii of two circles are 12 cm and 5 cm.
Distance of two central point is 20 cm
Distance between two central points
Suppose
Radius of fist circle = r1= 12 cm
Radius of second circle = r2 = 5 cm
Distance between two center point = s = 20 cm
Length of outer common tangents = t
0.5
0.5
1
Length of outer common tangent is square root of difference of square of
distance two center point of circle and square of difference of two radii of
circle
1
1
Its mean
=
=
=
−( − )
20 − ( 12 − 5 )
20 − ( 7 )
1
1
1
= √400 − 49
= √351
t = 18.7
1
1
Fifth meeting
Phase 1
E xploration
Student remember all concept of tangents of circle
Property of tangents of circle
1. E ach tangent perpendicular to one diameter or one radius of circle
2. Pass a point on circle, only a tangent can be created
3. Pass a point outside of circle, can create two tangents.
L ength of tangent of circle is square root of difference of square distance of center to a point outside the circle and square
of radius of that circle
Position of two circle
No
1.
Picture
r2
r1
Position
Intersection
Caracteristic
r1 + r2 > s
Contiguous
outside
r1 + r2 = s
Contiguous
inside
r1 + r2 > s
Disjoint outside
r1 + r2 < s
Concentric
r1+ r2= 0
s
2.
r1
r2
s
3.
r1
s
r2
4.
r1
s
5.
r1
r2
M
N
r2
15’
6.
Disjoint inside
r1 + r2 > s
s
r1
r2
There are 2 inner common tangents if two circles are disjoint outside
L ength of inner common tangents is square root of difference of square of distance two center point of circle and square of
addition of two radii of circle
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
Phase 7
Phase 8
Phase 9
There are 2 outer common tangents if two circles are disjoint outside
L ength of outer common tangents is square root of difference of square of distance two center point of circle and square of
difference of two radii of circle
Student is given double homework because no body collect it on time.
15’
Student discuss their homework no 7
Dua buah lingk aran bersinggungan di luar. L ingk aran besar memilik i jari-jari x cm dan lingk aran yang lainnya
berjari-jari 7 cm. Jik a panjang garis singgung persek utuan luarnya 30 cm, tentuk an panjang x dan luk is garis
singgung persek utuan luar tersebut!
Student prepares themselves for studying. Student knows learning objectives of lesson today
The students are able to:
1. Using concept of tangent of circle on solving the problem
2. Understanding the minimum rope relate two circles
Student mention some realistic example of minimum rope relate two circles
Bundle of PV C, bundle of wood
E laboration
Student discuss how to determine minimum rope related based on problem given
Student concludes how to determine minimum rope related.
Student determines the minimum rope related by joint all central points of the object.
Student solve the problem on Erlangga book page
Confirmation
Student discuss the problem 76-77
Phase 10 Student conclude lesson today and know their homework
Solving the minimum rope related by sketching the problem, and then joint each central points
and make parallel line to the outer side.
Assessment
: assessment of homework
For each problem:
1. Student gets 3 points if they find given and ask from the problem
2. Student gets 7 points if their answer correct
5’
5’
10’
5’
20’
10’
5’
School
: ………………………..
Subject
: Mathematics
Grade/ semester
: VIII/ II
Number of Meeting : 5 Meetings
Year of Lesson
: 2011/ 2012
Standard of Competency
Deter mine elements and par ts of cir cle and also measur ements
Basic Competence
Deter mine length of tangent of tw o cir cles
Indicators
1. Sketch the gr aph of tangent pass a point and find the pr oper ty of tangent of cir cle
2. Deter mining length of tangent of cir cle
3. Under standing position of tw o cir cles
4. Under standing and sketch inter nal and exter nal common tangents of cir cles
5. Deter mining length of inter nal and exter nal common tangents of cir cle
6. Deter mining the minimum r ope r elate tw o cir cles
Learning Objectives
:
After studying, students ar e able to:
1. Sketch the gr aph of tangent pass a point lie on cir cle and find the proper ty of tangent
2. Sketch the gr aph of tangent pass a point exter nal of cir cle and find the pr oper ty of
tangent
3. Deter mine length of tangent of cir cle
4. Under stand position of two cir cles and identify the cr iter ia for each position
5. Solve applied problem about length of tangent and position of two cir cles
6. Under stand and sketch inter nal and exter nal common tangents of cir cles
7. Deter mine length of inter nal and exter nal common tangents of cir cle
8. Solve applied problem about inter nal and external common tangents
9. Deter mine the minimum rope r elate tw o cir cles or mor e
Prerequisite materials
:
Pythagorean theorems
Materials
:
tangent of cir cle
Headline of t opic
1. Tangent of cir cle int er sect t he cir cle at one and only one point on t he cir cle.
2. Pr oper t y of t angent of cir cle:
a. Each t angent per pendicular t o diamet er or r adius of cir cle
b. Pass a point on cir cle, only a t angent can be cr eat ed
c. Pass a point out side of cir cle, can cr eat e t wo t angent s.
3. Lengt h of t angent of cir cle is squar e r oot of differ ence of squar e dist ance of cent er t o a
point out side t he cir cle and squar e of r adius of t hat cir cle
4. Lengt h of int er nal common t angent is squar e r oot of differ ence of dist ance of t wo
cir cles and squar e r oot of sum of r adii.
5. Lengt h of ext er nal common t angent is squar e r oot of differ ence of squar e of dist ance
t wo cent er point of cir cle and squar e of differ ence of t wo r adii of cir cle
Time Allocations
:
Method/ Approach/ Model
Learning resources
11 x 40 minutes
:
demonstr ation, drill and pr actice, constr uctivism, RME,
PBI
:
Learning Activities
First meeting
Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
Phase 7
Phase 8
Phase 9
Phase 10
Phase 11
Student prepares themselves to study. Teacher says learning
objectives and ask some realistic problem of circle’s tangent
Realistic problem of tangent of circle are bicycle’s chain,
pulley, bulldozer’s wheel, etc.
Exploration
Student remember definition of tangent
Tangent is line which is intersects circle exact one point
on the circle.
Student translate how to sketch tangent pass a point on a circle
Student follow teacher explanation about sketching tangent pass a
point on the circle
Sketch tangent pass a point on a circle
Elaboration
Student does exercise page 238 latihan 1 no 1 and 2
Student translate how to sketch tangent pass a point outside the
circle
Student follow teacher explanation about sketching tangent pass a
point outside the circle
Student do exercise page 238 latihan 1 no 3 and 4
Confirmation
Conclude the property of tangent of a circle
Property of tangent of circle:
1. Each tangent perpendicular to one diameter or one radius of
circle
2. Pass a point on circle, only a tangent can be created
3. Pass a point outside of circle, can create two tangents.
Conclude today lesson
Student is given homework
Rubric for exercise
Problem
Indicators
1
Have an idea to sketch tangent
Do the idea
Finding property of tangent based on him/her picture
5 minutes
5 minutes
5 minutes
10 minutes
10 minutes
5 minutes
10 minutes
10 minutes
5 minutes
5 minutes
5 minutes
1 2 3 4
2
Have an idea to sketch tangent
Do the idea
Finding property of tangent based on him/her picture
Rubric for problem sheet
Problem
Indicators
1
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
2
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
3
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
4
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
1= achieve the indicator less than 25%
2= achieve indicator between 25%-50%
3=achieve indicator indicators between 50%-75%
4= achieve indicator more than 75%
1 2 3 4
St udent ’s Problem Sheet
NAME
:
…………..
CLASS
:
………….
LENGTH OF TANGENT OF
CIRCLE
Standard Competence
Determine elements and parts of circle and also measurement
Basic Competence
Determine length of tangent of two circles
Indicator
Determine length of tangent of a circle
Direction!
1. Read summary topic first!
2. Answer exercise based on structure problem solving steps that your teacher explains!
Summary of Topic
Tangent of circle is line which is intersects circle on one point at the circle.
Length of tangent of circle is square root of difference of square distance of center to a point
outside the circle and square of radius of that circle
Exercise
1. Length of tangent of a circle is 24 cm, distance between center to a point outside of circle is
26 cm. Determine radius of circle!
Solve using structure problem solving steps
Given
Ask
Solve
2. Look at the picture!
Given radius of a circle, r = 6 cm and length AB = 10 cm.
Determine length of tangent of circle and find another
tangent of circle pass point B!
Solve using structure problem solving steps
Given
Ask
Solve
3. Determine the area of triangle POQ based on the following picture!
25 cm
O
Q
24 cm
P
Solve using structure problem solving steps
Given
Ask
Solve
O
B
A
4. Two woods which are look like a circle is bundled with 144 cm rope. If radii of two roots are same,
determine the radius!
Given
Ask
Solve
Second Meeting
Phase 1
Student discuss their homework at Erlangga book page 61 no 1-5
Phase 2
Student prepares themselves for studying. Student knows learning
objectives of lesson today
Understanding position of two circles
Phase 3
Exploration
Student make figure of some position of two circle based on
Erlangga book page 62
No
1.
Picture
r2
r1
s
2
r1
.
3.
r1
r2
s
r2
s
15 minutes
5 minutes
20 minutes
4.
r1
r2
s
5.
r1
r2
M
N
6.
s
r1
r2
Phase 4
Phase 5
Student pays attention to teacher’s explanation. Teacher show
some realistic condition of position of two circles and guide
student to conclude position and characteristic of each example
that is given.
1. Position of coin Rp 100 and Rp 500 if it is patched
over.
2. Position of minute hand and hours hand of watch
Elaboration
Student give the name of each position and also its characteristic
No
1.
Picture
r2
r1
Position
Intersection
Caracteristic
r1 + r2 > s
Contiguous
outside
r1 + r2 = s
Contiguous
inside
r1 + r2 > s
s
2.
r1
3.
r1
r2
s
r2
s
5 minutes
30 minutes
4.
r1
s
r2
5.
Disjoint
outside
r1 + r2 < s
Concentric
r 1+r 2=0
Disjoint
inside
r1 + r2 > s
r1
r2
M
N
6.
s
r1
r2
Student do the problem sheet
Confirmation
Discuss the problem sheet
Phase 8
Student know their homework to translate how to sketch internal
and external common tangents
Assessment :
problem sheet
10 minutes
5 minutes
Phase 6
Phase 7
5 minutes
Third Meeting
Phase 1
Phase 2
Phase 3
Phase 4
Student review their knowledge about position of two circle
Show in power point
Student prepares themselves for studying. Student knows learning objectives of lesson today
The students are able to:
1. Understanding internal and external common tangents
2. Sk etching internal and external common tangents
5’
Student translate how to sketch internal common tangents
10’
25’
E xploration
Student sketching internal common tangents
No
Steps
1.
Sketch a circle where center is P
and the radius is r1, and then circle
with central point is Q and radius
is r2.
Picture
r1
P
Q
r2
5’
2.
Join center of both circle
Q
P
3.
Sketch arc of circle with center
point P and point Q with same
radius. Both of arc intersect on
point A and B.
A
P
Q
B
4.
Join point A and point B, so that
its intersect line PQ on point C.
A
Q
P
C
B
5.
Sketch a circle with center point is
point C and diameter CP.
A
P
Q
C
B
6.
7.
Make an arc with radius r1+ r2 and
central point is point P at the top
and at the bootom.
Intersection of these arc are H
and I. Joint point P and H and
also point P and I. Intersection of
line PH and circle P is point D.
Intersection of line PI and circle P
is point E .
D
A
P
F
Q
C
G
B
E
Make arc to circle Q with radius
DQ and the central points are D
D
A
F
Q
P
C
B
E
G
8.
Joint point D and G and also
point E and point F.
Line DG and E F are internal
common tangents.
D
F
A
Q
P
C
G
B
E
Phase 5
Phase 6
Student mention some realistic example of external common tangents
Student sketching external common tangent
No
1.
2.
Steps
Sketch circle with center point is
point P and radius r1, then circle
where center is Q and radius r2.
20’
Picture
P
r1
Q
Join both center of that circle.
P
3.
Sketch arc of circle with center
point on point P and point Q and
have same radius. Both arc
intersect on point A and point B.
Q
A
P
Q
B
4.
Join point A and point B, so line
AB intersect line PQ on point C.
A
P
C
Q
B
5.
r2
Sketch a circle where center is C
and radius CP.
A
P
C
Q
B
6.
Sketch a circle with center point is
point P and radius r1-r2. This circle
intersect circle with central point
is point C on point D and point
E.
A
D
C
P
Q
E
B
7.
Join point P and point D so that
intersect circle with cetral point is
point P and radius r1 on point M.
After that join point P and point
E , so that intersect circle with
center point is point P with radius
r1 on point N.
Name point K and point L for
intersection of circle with the
center is point C and point Q.
Join point M and point K, and
then join point N and point L.
Line KM and line LN are external
common tangent of circle with
center points are point P and
point Q.
8.
M
A
D
C
P
Q
E
L
B
N
M
D
P
A
K
C
Q
E
L
B
N
Phase 7
K
E laboration
10’
Students try to sketch internal and external common tangent for another circles
Phase 8
Confirmation
5’
Students conclude the lesson
Performance Assessment
1. Internal common tangents
No
1.
Steps
Sketch a circle where center is P
and the radius is r1, and then circle
with central point is Q and radius
is r2.
Picture
r1
P
Q
0 1 2
r2
2.
Join center of both circle
Q
P
3.
Sketch arc of circle with center
point P and point Q with same
radius. Both of arc intersect on
point A and B.
A
P
Q
B
4.
Join point A and point B, so that
its intersect line PQ on point C.
A
P
Q
C
B
5.
Sketch a circle with center point is
point C and diameter CP.
A
P
Q
C
B
6.
Make an arc with radius r1+ r2 and ```````````````````````````````````````````````````````
central point is point P at the top ```````````````````````````````````````````````
D
and at the bootom.
A F
Intersection of these arc are H
Q
P
C
and I. Joint point P and H and
also point P and I. Intersection of
B
G
line PH and circle P is point D.
E
Intersection of line PI and circle P
is point E .
7.
Make arc to circle Q with radius
DQ and the central points are D
D
A
F
Q
P
C
B
E
G
8.
Joint point D and G and also
point E and point F.
Line DG and E F are internal
common tangents.
D
F
A
Q
P
C
G
B
E
2. External common tangents
No
1.
2.
Steps
Sketch circle with center point is
point P and radius r1, then circle
where center is Q and radius r2.
0 1 2
Picture
P
r1
Q
Join both center of that circle.
P
3.
Sketch arc of circle with center
point on point P and point Q and
have same radius. Both arc
intersect on point A and point B.
Q
A
P
Q
B
4.
Join point A and point B, so line
AB intersect line PQ on point C.
A
P
C
Q
B
5.
Sketch a circle where center is C
and radius CP.
A
C
P
Q
B
6.
r2
Sketch a circle with center point is
point P and radius r1-r2. This circle
A
D
P
C
Q
E
B
intersect circle with central point
is point C on point D and point
E.
7.
8.
Join point P and point D so that
intersect circle with cetral point is
point P and radius r1 on point M.
After that join point P and point
E , so that intersect circle with
center point is point P with radius
r1 on point N.
Name point K and point L for
intersection of circle with the
center is point C and point Q.
Join point M and point K, and
then join point N and point L.
Line KM and line LN are external
common tangent of circle with
center points are point P and
point Q.
M
A
D
K
C
P
Q
E
L
B
N
M
D
P
A
C
K
Q
E
N
B
L
Note:
0= student don’t do the step
1= student do the step but not suitable
2= student do the step correct
Forth meeting
Phase
Activities
1.
Students pr epar e themselves to study
2.
Explor ation
Students mention some object w hich r elated to inter nal and exter nal
common tangents
3.
Students know the lear ning objectives of the lesson and the r ule of
lesson for today
4.
Students r emind how to sketch inter nal and exter nal common tangent
One of the students sketch the tangents on w hite/ blackboar d
5.
Elabor ation
Teacher br ing students to find r ight triangle on inter nal common
tangent and dir ect students to use the concept of Pythagor ean
theor em to deter mine the length of inner common tangents
Time
5’
5’
5’
5’
6.
7.
8.
9.
By using same w ay, students find the length of exter nal common
tangents
Students solve some pr oblem about inter nal and exter nal common
tangents
Teacher help students to solve their pr oblem
Confir mation
Students discuss the most difficulties pr oblem on class discussion
Student conclude lesson today and know their homework
There are 2 outer common tangents if two circles are disjoint outside
L ength of outer common tangent is square root of difference of square of distance two center
point of circle and square of difference of two radii of circle
Assessment of exercise
1. Radii of two circles are 20 cm and 4 cm. determine the distance between two central points
if known length of outer common tangents is 26 cm.
(1 point)
Given
Radii of two circles are 20 cm and 4 cm.
0.5
Length of outer common tangents is 26 cm
0.5
Ask
Distance between two central points
1
Solve
Suppose
Radius of fist circle = r1= 20 cm
Radius of second circle = r2 = 4 cm
Length of outer common tangents = t = 26 cm
Distance between two center point = s
Length of outer common tangent is square root of difference of square of
distance two center point of circle and square of difference of two radii of
circle
Its mean
−( − )
26 =
− ( 20 − 4 )
26 =
− ( 16 )
2
2
26 =
− 16 2
=
2
2
2
= 26 + 16
= 676 + 256
2
= 932
= √932
s = 30.5
1
1
1
1
2
1
1
1
2. Given radii of two circles are 16 cm and 17 cm. Distance of two central points is 24 cm.
Calculate the length of outer common tangents!
(1point)
Given
Radii of two circles are 16 cm and 17 cm.
0.5
Distance of two central point is 24 cm
0.5
Ask
Distance between two central points
1
Solve
Suppose
Radius of fist circle = r1= 16 cm
Radius of second circle = r2 = 17 cm
1
Distance between two center point = s = 24 cm
Length of outer common tangents = t
Length of outer common tangent is square root of difference of square of
distance two center point of circle and square of difference of two radii of
circle
Its mean
=
=
−( − )
24 − ( 16 − 17 )
24 − ( 1 )
√576 − 1
√575
=
=
=
t = 23.9
1
1
1
1
1
1
3. Suppose radii of two circles are 12 cm and 5 cm. Distance of two central points is 20 cm.
Calculate the length of outer common tangents!
(1 point)
Given
Ask
Solve
Radii of two circles are 12 cm and 5 cm.
Distance of two central point is 20 cm
Distance between two central points
Suppose
Radius of fist circle = r1= 12 cm
Radius of second circle = r2 = 5 cm
Distance between two center point = s = 20 cm
Length of outer common tangents = t
0.5
0.5
1
Length of outer common tangent is square root of difference of square of
distance two center point of circle and square of difference of two radii of
circle
1
1
Its mean
=
=
=
−( − )
20 − ( 12 − 5 )
20 − ( 7 )
1
1
1
= √400 − 49
= √351
t = 18.7
1
1
Fifth meeting
Phase 1
E xploration
Student remember all concept of tangents of circle
Property of tangents of circle
1. E ach tangent perpendicular to one diameter or one radius of circle
2. Pass a point on circle, only a tangent can be created
3. Pass a point outside of circle, can create two tangents.
L ength of tangent of circle is square root of difference of square distance of center to a point outside the circle and square
of radius of that circle
Position of two circle
No
1.
Picture
r2
r1
Position
Intersection
Caracteristic
r1 + r2 > s
Contiguous
outside
r1 + r2 = s
Contiguous
inside
r1 + r2 > s
Disjoint outside
r1 + r2 < s
Concentric
r1+ r2= 0
s
2.
r1
r2
s
3.
r1
s
r2
4.
r1
s
5.
r1
r2
M
N
r2
15’
6.
Disjoint inside
r1 + r2 > s
s
r1
r2
There are 2 inner common tangents if two circles are disjoint outside
L ength of inner common tangents is square root of difference of square of distance two center point of circle and square of
addition of two radii of circle
Phase 2
Phase 3
Phase 4
Phase 5
Phase 6
Phase 7
Phase 8
Phase 9
There are 2 outer common tangents if two circles are disjoint outside
L ength of outer common tangents is square root of difference of square of distance two center point of circle and square of
difference of two radii of circle
Student is given double homework because no body collect it on time.
15’
Student discuss their homework no 7
Dua buah lingk aran bersinggungan di luar. L ingk aran besar memilik i jari-jari x cm dan lingk aran yang lainnya
berjari-jari 7 cm. Jik a panjang garis singgung persek utuan luarnya 30 cm, tentuk an panjang x dan luk is garis
singgung persek utuan luar tersebut!
Student prepares themselves for studying. Student knows learning objectives of lesson today
The students are able to:
1. Using concept of tangent of circle on solving the problem
2. Understanding the minimum rope relate two circles
Student mention some realistic example of minimum rope relate two circles
Bundle of PV C, bundle of wood
E laboration
Student discuss how to determine minimum rope related based on problem given
Student concludes how to determine minimum rope related.
Student determines the minimum rope related by joint all central points of the object.
Student solve the problem on Erlangga book page
Confirmation
Student discuss the problem 76-77
Phase 10 Student conclude lesson today and know their homework
Solving the minimum rope related by sketching the problem, and then joint each central points
and make parallel line to the outer side.
Assessment
: assessment of homework
For each problem:
1. Student gets 3 points if they find given and ask from the problem
2. Student gets 7 points if their answer correct
5’
5’
10’
5’
20’
10’
5’