Right-Angled Triangle Check: Is ABC A Right Triangle?

by Alex Johnson 54 views

Let's dive into the fascinating world of triangles and explore whether a specific triangle, ABC, qualifies as a right-angled triangle. This article will guide you through the process of determining this, given the perimeter and the lengths of two sides. We'll use clear, step-by-step calculations to arrive at our conclusion. So, grab your thinking cap, and let's get started!

Understanding Right-Angled Triangles and the Pythagorean Theorem

Before we jump into the specific problem, it's crucial to understand what defines a right-angled triangle. A right-angled triangle, also known as a right triangle, is a triangle that has one angle exactly equal to 90 degrees. This 90-degree angle is often marked with a small square in the corner where the two sides meet.

The most important tool for determining if a triangle is right-angled is the Pythagorean Theorem. This theorem states that in a right-angled triangle, the square of the length of the longest side (called the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides (called the legs). Mathematically, this is expressed as: a² + b² = c², where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the other two sides.

The Pythagorean Theorem is a fundamental concept in geometry and is used extensively in various fields, including engineering, architecture, and navigation. It provides a simple yet powerful way to verify if a triangle adheres to the properties of a right-angled triangle. Knowing this theorem is the key to solving our problem.

To effectively utilize the Pythagorean Theorem, it's imperative to accurately identify the longest side within the triangle, which corresponds to the hypotenuse. The hypotenuse is invariably situated opposite the right angle, and it is the cornerstone for verifying the theorem's applicability. Once we have determined all three sides of our triangle, we can plug them into the equation a² + b² = c² and check if the equation holds true. If it does, then we can confidently conclude that we are dealing with a right-angled triangle. Conversely, if the equation does not hold true, it signifies that the triangle is not a right-angled triangle. This meticulous approach, grounded in the principles of the Pythagorean Theorem, empowers us to make precise determinations regarding the nature of triangles and their geometrical properties.

Problem Setup: Triangle ABC and Its Properties

Now, let's focus on the specific triangle we're dealing with: triangle ABC. We are given the following information:

  • The perimeter of triangle ABC is 22 cm.
  • Side AB has a length of 8 cm.
  • Side BC has a length of 5 cm.

Our goal is to determine if triangle ABC is a right-angled triangle. To do this, we need to find the length of the third side, AC, and then apply the Pythagorean Theorem.

The perimeter of a triangle is the total length of all its sides combined. Therefore, we can express the perimeter of triangle ABC as: AB + BC + AC = 22 cm. We already know the lengths of AB and BC, so we can substitute those values into the equation: 8 cm + 5 cm + AC = 22 cm. This equation allows us to solve for the unknown side, AC, and complete the information we need to apply the Pythagorean Theorem. Once we find the length of AC, we can compare the squares of the sides to see if the relationship a² + b² = c² holds true, thereby confirming or denying that triangle ABC is a right-angled triangle.

Calculating the Length of Side AC

To find the length of side AC, we'll use the perimeter information provided. As we established earlier, the perimeter is the sum of all the sides:

AB + BC + AC = 22 cm

We know AB = 8 cm and BC = 5 cm. Substituting these values, we get:

8 cm + 5 cm + AC = 22 cm

Combining the known values:

13 cm + AC = 22 cm

Now, to isolate AC, we subtract 13 cm from both sides of the equation:

AC = 22 cm - 13 cm

AC = 9 cm

So, the length of side AC is 9 cm. With the lengths of all three sides now known (AB = 8 cm, BC = 5 cm, and AC = 9 cm), we can move forward to the crucial step of applying the Pythagorean Theorem to ascertain whether triangle ABC is indeed a right-angled triangle. This involves comparing the sum of the squares of the two shorter sides with the square of the longest side, allowing us to definitively conclude the triangle's nature.

Applying the Pythagorean Theorem

Now that we know all three sides of the triangle (AB = 8 cm, BC = 5 cm, and AC = 9 cm), we can use the Pythagorean Theorem to check if it's a right-angled triangle. Remember, the theorem states: a² + b² = c², where 'c' is the length of the hypotenuse (the longest side).

In our case, the longest side is AC, which is 9 cm. So, we'll consider AC as 'c', and AB and BC as 'a' and 'b' respectively. Let's plug the values into the equation:

8² + 5² = 9²

Now, we calculate the squares:

64 + 25 = 81

Adding the numbers on the left side:

89 = 81

As you can see, 89 is not equal to 81. This means that the equation a² + b² = c² does not hold true for triangle ABC. Therefore, based on the Pythagorean Theorem, we can conclude that triangle ABC is not a right-angled triangle.

Conclusion: Triangle ABC is Not Right-Angled

Through our step-by-step calculations, we have determined that triangle ABC, with sides AB = 8 cm, BC = 5 cm, and AC = 9 cm, and a perimeter of 22 cm, is not a right-angled triangle. We arrived at this conclusion by first calculating the length of the missing side, AC, using the perimeter information. Then, we applied the Pythagorean Theorem, a fundamental principle for identifying right-angled triangles. Since the equation a² + b² = c² did not hold true when we substituted the side lengths, we definitively established that triangle ABC does not possess a 90-degree angle.

Understanding and applying the Pythagorean Theorem is a crucial skill in geometry and is essential for solving various problems involving triangles and their properties. This example demonstrates how to use the theorem to classify triangles and reinforces the importance of accurate calculations and logical reasoning in mathematics.

For further exploration of triangles and the Pythagorean Theorem, you can visit Khan Academy's Geometry section.