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Name Gender
Address Phone E-mail
Designers Caswita M Lampung (0721)267529 [email protected]
Directors Herawati F Jakarta (021) 7414952 [email protected]
Teacher Mery F Jakarta 081319204959 [email protected]
Prod. Of coursware
Isa &
Abdul kadir
M
M
Semarang
Malang
081326527310 [email protected]
Prod. Of CD room
Wayan M Bali (0362)31955 [email protected]
Target state :
Students can be able to draw common tangent by using a compass and a ruler as well the length of it
Student Objective
1. identifying the position of two circles and re-cognizing the common tangent of them (if it exists)
2. drawing and calculating the length of common tangents of circles
3. solving problems involving tangents
4. explaining tangents kites
Content analysis
Keypoint : common tangent of circles Topic student difficult to understand: drawing
tangent of circles by compass and determine the length of tangent
Relationship of content to be learnt and the other content (circles, line, phytagoras theorm)
Relationship between different topics within the content area: position of the circles, drawing tangent by compass and determining length of tangent
Student analisis
Cognitive analysisDrawing and calculating length of tangent
Prior analisis: students know about lines, circles, Pythagoras theorm
skill of student: they can use compass and ruler
Procedure and teaching strategies
Direct learning Cooperative Learning
Description of teaching and learning processPhases Teaching
ActivitiesLearning Act Ass. alongside Use of ICT Time needed
Intro Teachers tell about the objective of learning
Student listen about the object
Teacher give a questions
Powerpoint 5
Main activities
Techers tach subject
Student solve problems in group and then present the result
Problems Powerpoint, compass
70
closing Techer give the summaries and homework
Student listen and ask questions
exercise powerpoint 15
8.2 Common tangent of circles
Objectives:
1. identifying the position of two circles and re-cognizing the common tangent of them (if it exists)
2. drawing and calculating the length of common tangents of circles
3. solving problems involving tangents
4. explaining tangents kites
The students are expected to be capable
The position of two circles
Given two circles centered at point M and N with radius R and r where R > r, respectively
Problem :
1. How many possibilities the position of two circles can be made ?
2. Determine the number of lines that touch each circle exactly at one point ?
The length of MN is less than (R - r)
Then the circle N is inside the circle M,
M
R N
r
M = N
Rr
On the Case M = N, the circles M and N are called concentric circles (the circles have a common center)
so thatthere is no line touching each circle exactly at one point
Case 1:
Case 2 : The length of MN is equal to (R – r).
so that there is only one line touching each circle exactly at one point so-called common tangent (of circles M and N)
M
RN
r
circle N touches inside the circle M exactly at one point,
Then the
Case 3 : The length of MN is equal to (R + r).
circle N touches outside the circle M exactly at one point,
Then the
so that there are three lines touching each circle exactly at one point
O
P
Q
The Line through O and perpendicular to line MN is called inner common tangent, the lines PQ and ST are called outer common tangent (of circles (M,R) and (N,r)
S
T
M
R
Nr
Work in Group :
Determine the number of common tangent of Circles (M ; R) and (N ; r) for the other cases, and show by pictures
Drawing common tangent of circles and determining the length of it
Given two circles centered M and N with radius R and r such that MN > R + r
Problem:
1. How to draw inner and outer common tangents ?
2. Determine the length of inner and outer common tangents ?
ProcedureDrawing inner common tangent
1. Draw circles (M ; R) and (N ; r) with MN > R+r
2. Draw a circle centered at M with radius of R+r
3. Draw a line passing points M and N, and then find mid-point O of it
4. Draw circle with center at O and radius OM, then it intersects circle (M ; R+r) at two points, namely P and Q
5. Draw lines NP and NQ, then they are tangent of circle (M ; R+r) ( Why ? )
6. Draw line through N downward and parallel to PM intersecting circle (N ; r) at S, then line ST paralel to NP is an inner common tangent
M
R
rN
O
P
Q
S
T
7. Similarly, the another inner common tangent UV will be found
U
V
The length of inner common tangent
means that distance of common point of tangent and one circle to that of tangent and another
Observe right-angled triangle NPM, then 22 )()( NPMNNP
Because ST // NP and NS // PT, then ST = NP so that the length of inner common tangent ST is equal to
22 )()( NPMN Similarly, the length of inner common tangent UV is equal to that of ST
Drawing outer common tangent
1. Draw circles (M ; R) and (N ; r) with MN > R+r
2. Draw a circle centered at M with radius of R-r
3. Draw a line passing points M and N, and then find mid-point of it, namely O
4. Draw circle with center at O and radius OM, then it intersects circle (M ; R-r) at two points, namely P and Q
5. Draw lines NP and NQ, then they are tangent of circle (M ; R-r) ( Why ? )
6. Draw line through N upward and parallel to PM intersecting circle (N ; r) at S, then line ST which is parallel to NP is an inner common tangent
M
R
rN
O
P
Q
S
T
U
V
7. Similarly, the another inner common tangent UV will be found
The length of inner common tangent
Observe right-angled triangle NPM, then
22 )()( NPMNNP Because ST // NP and NS // PT, then ST = NP so that the length of inner common tangent ST is equal to
22 )()( NPMN Similarly, the length of inner common tangent UV is equal to that of ST
Work in Group :
Draw common tangent of Circles (M ; R) and (N ; r) for the case MN = R+r, and determine the length of them