Summary about the thales theorem for the middle school

1. Thales Theorem
Chapter 8-Constructions

2. Applications:
-Construction of the fourth
proportional
-Dividing a segment into equal
parts.
-Constructions of points
-Enlargement –Reduction

3. Construction of the fourth proportional
• Question :construct the fourth proportional to 4,7 and 5.
Etape 1:Trace a straight line (d) and place 3 points O,A & B in the following order :OA = 4cm and OB =7cm.
O A B (d)
0 4 7
Etape 2:Trace a straight line (d′) that cuts (d) at O and place the point C such that OC = 5cm.
O A B (d)
0 4 7
C
5
D
x (d′)

4. Step 3: construct the parallel to (AC) passing through B, it cuts (d’) at D.
According to Thales’ theorem, the length x of segment [OD] is: 5/x = 4/7 or x x 4 = 5 x 7
The length x of segment [OD] is the fourth proportional to 4,7 and 5.

5. Dividing a segment into equal parts

6. Dividing a segment into equal parts
Divide [AB] into 4 equal parts

7. Construction of points
Place point K on segment [AB] such that 𝐴M =
2
5
𝐴𝐵 OR
AM
AB
=
2
5

12. Construction of points
Place points M on line (AB) such that
𝑀𝐴 =
2
5
𝑀𝐵 OR
𝑀𝐴
𝑀𝐵
=
2
5

13. Construction of points

14. Construction of points

15. Construction of points

16. Construction of points

17. Construction of points
Place points M on line (AB) such that 𝑀𝐴 =
2
3
𝑀𝐵 OR
𝑀𝐴
𝑀𝐵
=
2
3

18. Construction of points
Place points M on line (AB) such that 𝑀𝐴 =
2
3
𝑀𝐵 OR
𝑀𝐴
𝑀𝐵
=
2
3

20. Construction of points
Place points M on line (AB) such that
𝑀𝐴
𝑀𝐵
=2

22. Enlargement and Reduction
Enlargement and reduction can be seen in many
situations in real life.

23. Elements of enlargement and reduction
The elements of enlargement or reduction are:
1) The ratio or the coefficient k:
𝑘 =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑛𝑒𝑤 𝑠𝑖𝑑𝑒 (𝑖𝑚𝑎𝑔𝑒)
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
• If k>1,then the image is an enlargement
• If k<1, then the image is reduction
2) Center of enlargement or reduction

26. Application:
Given segment AB and A’B’ the image of AB by enlargement
A
B
A’
B’
1) Find the center of enlargement O
2)Suppose that OA’=12 and OA=3
Find the ratio of enlargement
O

27. Enlarge triangle ABC of ratio 3 and center O
𝑂𝐵′
𝑂𝐵
= 3 𝑡ℎ𝑒𝑛 𝑂𝐵′
= 3𝑂𝐵
𝑂𝐶′
𝑂𝐶
= 3 𝑡ℎ𝑒𝑛 𝑂𝐶′ = 3𝑂𝐶
𝑂𝐴′
𝑂𝐴
= 3 𝑡ℎ𝑒𝑛 𝑂𝐴′ = 3𝑂𝐴

28. Enlarge triangle ABC of ratio 3 and center A
𝐴𝐵′
𝐴𝐵
= 3 𝑡ℎ𝑒𝑛 𝐴𝐵′
= 3𝐴𝐵
𝐴𝐶′
𝐴𝐶
= 3 𝑡ℎ𝑒𝑛 𝐴𝐶′ = 3𝐴𝐶
𝐴𝐴′
𝐴𝐴
= 3 𝑡ℎ𝑒𝑛 𝐴𝐴′
= 3𝐴𝐴
But AA=0 then AA’=0
This means that A and A’ are
confounded

29. Reduce triangle ABC of ratio 1/2 and center O
𝑂𝐵′
𝑂𝐵
=
1
2
𝑡ℎ𝑒𝑛 𝑂𝐵′
=
1
2
𝑂𝐵
𝑂𝐶′
𝑂𝐶
=
1
2
𝑡ℎ𝑒𝑛 𝑂𝐶′ =
1
2
𝑂𝐶
𝑂𝐴′
𝑂𝐴
=
1
2
𝑡ℎ𝑒𝑛 𝑂𝐴′ =
1
2
𝑂𝐴

30. Reduce triangle ABC of ratio 1/2 and center A
𝐴𝐵′
𝐴𝐵
=
1
2
𝑡ℎ𝑒𝑛 𝐴𝐵′
=
1
2
𝐴𝐵
𝐴𝐶′
𝐴𝐶
=
1
2
𝑡ℎ𝑒𝑛 𝐴𝐶′ =
1
2
𝐴𝐶
𝐴𝐴′
𝐴𝐴
=
1
2
𝑡ℎ𝑒𝑛 𝐴𝐴′
=
1
2
𝐴𝐴
But AA=0 then AA’=0
This means that A and A’ are
confounded