Can 5 cm, 4 cm, and 10 cm Form a Triangle?

To determine whether three given lengths can form a triangle, we apply the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Let's apply this theorem to the given lengths: 5 cm, 4 cm, and 10 cm.

According to the triangle inequality theorem, the following conditions must be satisfied:

a b c b a c c a b

For the given lengths:

5 4 10 14 14 - True 4 5 10 14 14 - True 10 5 4 9 9 - False

Since the first condition is false (10 5 4 9 9 is false), the lengths 5 cm, 4 cm, and 10 cm cannot form a triangle. Therefore, the answer is no.

However, in a more nuanced perspective, we can consider the concept of a degenerate triangle or straight line segment.

A degenerate triangle can occur when one side of the triangle is equal to the sum of the other two sides. In this case, we have a situation where the 10 cm side is the sum of the 4 cm and 5 cm sides. While the lengths can technically form a figure, it collapses into a straight line segment due to the equality of the sum of the two shorter sides.

Mathematically, we can represent this scenario as follows:

- 5 4 10

This implies that a straight line segment can be formed with these lengths. In such a case, the interior angles of the "triangle" would be 0°, 0°, and 180°, making it a degenerate triangle.

Considering the Law of Cosines, it correctly calculates the angles for such degenerate cases, validating the formation of this geometric figure.

However, whether to include such a degenerate triangle in the class of triangles is often a matter of context and application.

For the second part of the question, since 5 6 11 and is greater than 10, a triangle can indeed be formed with sides of 5 cm, 6 cm, and 10 cm.

Here’s how to construct such a triangle:

Draw a line segment AB of 10 cm length. With point A as the center, draw a circle (or arc) with a radius of 5 cm. With point B as the center, draw another circle (or arc) with a radius of 6 cm. The intersection point of these two arcs is point C. Connect points A and C, and points B and C to form the triangle ABC.

By following these steps, you will have successfully constructed a triangle with sides of 5 cm, 6 cm, and 10 cm.