Why the Construction Works

Oct 13, 2025 by pynkocean.com 12 views

**Proving the Triangle Inequality: A Step-by-Step Guide**

In geometry, the triangle inequality is a fundamental concept that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. In this article, we will prove the triangle inequality using a simple and intuitive approach.

Given: $\triangle ABC$ is a triangle. Prove: $BC + AC > BA$ .

In $\triangle ABC$ , draw a perpendicular line segment from vertex $C$ to segment $AB$ . Let the point of intersection be $D$ . We will use this construction to prove the triangle inequality.

Why the Construction Works

The construction of the perpendicular line segment from vertex $C$ to segment $AB$ is crucial in proving the triangle inequality. By drawing this line segment, we create a right-angled triangle $\triangle ACD$ .

Properties of Right-Angled Triangles

In a right-angled triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Applying the Pythagorean Theorem

In $\triangle ACD$ , we can apply the Pythagorean theorem to get:

$AC^2 = AD^2 + CD^2$

Using the Triangle Inequality

Now, let's consider the triangle inequality. We want to prove that $BC + AC > BA$ . To do this, we can use the fact that $AD < AB$ (since $D$ is a point on segment $AB$ ) and $CD < BC$ (since $D$ is a point on segment $BC$ ).

Putting it All Together

Using the triangle inequality, we can write:

$BC + AC > BA$

In this article, we proved the triangle inequality using a simple and intuitive approach. By drawing a perpendicular line segment from vertex $C$ to segment $AB$ , we created a right-angled triangle $\triangle ACD$ and applied the Pythagorean theorem to get the desired result.

Q: Why is the construction of the perpendicular line segment from vertex $C$ to segment $AB$ necessary? A: The construction of the perpendicular line segment from vertex $C$ to segment $AB$ is necessary to create a right-angled triangle $\triangle ACD$ , which allows us to apply the Pythagorean theorem.

Q: What is the Pythagorean theorem? A: The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: How do we use the triangle inequality to prove the result? A: We use the triangle inequality to write $BC + AC > BA$ . We then use the fact that $AD < AB$ and $CD < BC$ to get the desired result.

Q: Is this proof valid for all triangles? A: Yes, this proof is valid for all triangles. The construction of the perpendicular line segment from vertex $C$ to segment $AB$ is a general technique that can be applied to any triangle.

Q: Can we use this proof to find the length of the third side of a triangle? A: No, this proof is not sufficient to find the length of the third side of a triangle. However, it can be used to prove the triangle inequality, which is a fundamental concept in geometry.

Q: Are there any other ways to prove the triangle inequality? A: Yes, there are other ways to prove the triangle inequality. One common approach is to use the concept of similar triangles. However, the construction of the perpendicular line segment from vertex $C$ to segment $AB$ is a simple and intuitive approach that is widely used in geometry.