Relation between the Lengths of the Sides of a Triangle
Activity One:
Consider the triangle above. Move the vertices around to get different triangle. Create three different triangles like that and fill up the values for one triangle below.
Check for different triangles.
| Side1 | Side2 | Side3 | Side1+Side2 | Side1-Side2 |
|---|---|---|---|---|
| AB = | BC = | CA = | AB+BC = | AB-BC= |
| BC = | CA= | AB= | BC+CA= | BC-CA= |
| CA= | AB= | BC= | CA+AB= | CA-AB= |
For different values of the 3 sides, what do you observe ? The sum of two sides is more than the third side.
Also, the difference of two sides is
Therefore, we conclude that:
Triangle Inequality Theorem:
This theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Mathematically, for a triangle with sides a, b and c, this means:
a + b > c and b + c > a and c + a > b
This principle is crucial for the existence of a triangle. If this condition is not met, a triangle cannot be formed.
We also find that:
Side Length Difference:
The difference between the lengths of any two sides of a triangle must be less than the length of the third side. This can be seen as a corollary to the Triangle Inequality Theorem.
Mathematically, it is represented as:
∣a−b∣ < c
∣b−c∣ < a
∣c−a∣ < b
This condition ensures that the two shorter sides are sufficiently long to meet and form a triangle with the third side.
Example 3: Is there a triangle whose sides have lengths 10.2 cm, 5.8 cm and 4.5 cm?
Solution:
Suppose such a triangle is possible. Then the sum of the lengths of any two sides would be
Is 4.5 + 5.8 > 10.2 ?
Is 5.8 + 10.2 > 4.5 ?
Is 10.2 + 4.5 > 5.8 ?
Therefore, the triangle is possible.
Example 4: The lengths of two sides of a triangle are 6 cm and 8 cm. Between which two numbers can length of the third side fall?
Solution:
We know that the sum of two sides of a triangle is always greater than the third.
Therefore, third side has to be
The third side is thus, less than 8 + 6 =
The side cannot be
The length of the third side could be any length greater than 2 and less than 14 cm.
Is the sum of any two angles of a triangle always greater than the third angle ?
Yes, the sum of any two angles of a triangle is always greater than the third angle.
This is a consequence of the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
This is because in any triangle, the sum of all three interior angles is always