Understanding the Type of Triangle with Side Lengths 9 cm, 12 cm, and 15 cm

Triangles can be classified based on their side lengths, which can provide important insights into their properties. In this article, we will explore a triangle with side lengths of 9 cm, 12 cm, and 15 cm, and determine its type using mathematical principles, specifically the Pythagorean theorem.

Introduction to Right-Angled Triangles

A right-angled triangle is a triangle where one angle measures 90 degrees. The long side, opposite the right angle, is called the hypotenuse. The Pythagorean theorem is a powerful tool to identify right-angled triangles: if the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides, then the triangle is right-angled.

Using the Pythagorean Theorem

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. We can apply this theorem to our triangle with side lengths 9 cm, 12 cm, and 15 cm.

Verification:

Let's check if the given triangle satisfies the Pythagorean theorem:

Identify the longest side (hypotenuse): 15 cm. Apply the Pythagorean theorem: (c^2 a^2 b^2). Substitute the values: (15^2 9^2 12^2). Calculate the squares: (225 81 144). Check the equation: (225 225).

Since the equation holds true, we can confirm that the given triangle is a right-angled triangle.

Classifying the Triangle

Now that we have confirmed it is a right-angled triangle, we can further classify it based on its side lengths.

Scalene Triangle

A scalene triangle is one where all sides are of different lengths. The given triangle has sides of lengths 9 cm, 12 cm, and 15 cm, which means it is a scalene triangle.

Conclusion:

Therefore, the triangle with side lengths 9 cm, 12 cm, and 15 cm is a right scalene triangle. This classification applies because it is a right-angled triangle with all three sides of different lengths.

Additional Insights: Pythagorean Triples

A Pythagorean triple consists of three positive integers (a), (b), and (c) that fit the equation (a^2 b^2 c^2). Our triangle with sides 9 cm, 12 cm, and 15 cm is indeed a Pythagorean triple. This is not just a coincidence but part of a pattern seen in many right-angled triangles.

Here is an example of another Pythagorean triple: 5, 12, 13. When we check, we see that (5^2 12^2 13^2), which confirms that these form a right-angled triangle as well.

Other examples include 3, 4, 5, where (3^2 4^2 5^2).

In conclusion, the triangle with side lengths 9 cm, 12 cm, and 15 cm is a right-angled scalene triangle. This classification is based on the Pythagorean theorem and the properties of side lengths.