1. Relationship among the Three sides of a Triangle

2. Relationship among the Three Sides of a Triangle
Do you think any three line segments can form a triangle?
Maybe…

3. Relationship among the Three Sides of a Triangle
Let’s try to form a triangle with three line segments of lengths 3 cm, 4 cm and 5 cm. A B C A B C
= 3 cm
= 4 cm
= 5 cm

4. Relationship among the Three Sides of a Triangle
We can see that
AB + BC
= 3 cm + 4 cm
= 7 cm
> 5 cm
i.e. AB + BC > CA A B C A B
= 3 cm B C
= 4 cm C A
= 5 cm
AB + BC > CA

5. Relationship among the Three Sides of a Triangle
How does the shape of △ABC change when the length of AB decreases? A B C
= 3 cm B C
= 4 cm C A
= 5 cm
AB + BC > CA

6. Relationship among the Three Sides of a Triangle
How does the shape of △ABC change when the length of AB decreases? A B C
= 3 cm
= 2.5 cm A B C B C
= 4 cm C A
= 5 cm
AB + BC > CA

7. Relationship among the Three Sides of a Triangle
How does the shape of △ABC change when the length of AB decreases?
= 2.5 cm A B C A B C
= 2 cm B C
= 4 cm C A
= 5 cm
AB + BC > CA

8. AB + BC is not longer than CA.
Relationship among the Three Sides of a Triangle
How does the shape of △ABC change when the length of AB decreases?
When AB is not longer that 1 cm, these 3 lines cannot form a triangle. A B
1 cm A B C
= 2 cm B B C
= 4 cm A C C A
= 5 cm
AB + BC is not longer than CA.
AB + BC > CA

9. From the previous page, we find that …
The sum of the lengths of any two sides of triangle is greater than the length of the third side.
i.e. In △ABC, A B C
AB + BC > AC ;
AB + BC > AC ;
BC + AC > AB ;
BC + AC > AB ;
AB + AC > BC.
AB + AC > BC.
This relationship is known as the triangle inequality.

10. For example, in △ABC, AB + BC = 14 cm > AC BC + AC = 18 cm > AB
AB + AC = 16 cm > BC A B C
8 cm
10 cm
6 cm
AB + BC = 14 cm > AC
BC + AC = 18 cm > AB
AB + AC = 16 cm > BC
Conversely, when the lengths of three line segments are given, if the sum of any two lengths is greater than the third one, then the three line segments can form a triangle.

11. Follow-up question
In each of the following, the lengths of three line segments are given. Determine whether the three line segments can form a triangle and explain your answer briefly.
(a) 3 cm, 4 cm, 6 cm (b) 7 cm, 7 cm, 15 cm
Solution
(a) ∵ The sum of the lengths of the two shorter line segments = (3 + 4) cm
= 7 cm
> 6 cm
∴ The three line segments can form a triangle.

12. Follow-up question (cont’d)
In each of the following, the lengths of three line segments are given. Determine whether the three line segments can form a triangle and explain your answer briefly.
(a) 3 cm, 4 cm, 6 cm (b) 7 cm, 7 cm, 15 cm
Solution
(b) ∵ The sum of the lengths of the two shorter line segments = (7 + 7) cm
= 14 cm
< 15 cm
∴ The three line segments cannot form a triangle.