How Many Triangles Can be Formed with Sides 6 cm, 8 cm, and 10 cm?

Introduction:
When given the lengths of three sides of a triangle—such as 6 cm, 8 cm, and 10 cm—how many distinct triangles can be formed? The answer might surprise you. In this article, we will explore the conditions that define triangles and explain why only one triangle can be formed with these specific side lengths.

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any three sides a, b, and c of a triangle, the following inequalities must hold true:

a b c a c b b c a

Using this theorem, we can verify whether a triangle can be formed with sides 6 cm, 8 cm, and 10 cm:

Verification Using the Triangle Inequality Theorem

Designate the sides as follows:

a 6 cm b 8 cm c 10 cm

Now, we check the inequalities:

6 8 10 → 14 10, which is true 6 10 8 → 16 8, which is true 8 10 6 → 18 6, which is true

Since all three conditions of the triangle inequality theorem are satisfied, a triangle can indeed be formed with these side lengths.

Unique Determination by Side Lengths

The SSS (Side-Side-Side) Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. In other words, knowing the exact lengths of the sides of a triangle uniquely determines the triangle.

Therefore, when we have sides of 6 cm, 8 cm, and 10 cm, and using the SSS postulate, we can conclude that only one unique triangle can be formed. This triangle is a scalene triangle because all three sides are of different lengths.

Conclusion

In summary, when given the lengths of 6 cm, 8 cm, and 10 cm, only one triangle can be formed. This triangle is unique due to the conditions set by the triangle inequality theorem and the SSS postulate. No matter how you draw this triangle, it will always maintain these exact side lengths and angles, making it one and only one triangular shape.

Now, let’s delve deeper into related concepts and FAQs:

Related Concepts

Triangle Inequality Theorem: The comprehensive theorem that ensures the sides of a triangle can form a valid triangle. SSS Postulate: The postulate that guarantees congruence of triangles based on the length of their sides. Scalene Triangle: A triangle with all sides of different lengths.

FAQs

Can multiple triangles be formed with the same side lengths?
No, according to the SSS postulate, only one unique triangle can be formed with given side lengths. Are all triangles formed with the same side lengths congruent?
Yes, they are congruent, meaning they have the same shape and size. What happens if one of the side lengths is changed slightly?
The triangle inequality theorem must still hold. If it doesn't, a triangle cannot be formed.