Which Best Explains Whether A Triangle With Side Lengths $5 \, \text{cm}, 13 \, \text{cm}$, And $12 \, \text{cm}$ Is A Right Triangle?A. The Triangle Is A Right Triangle Because $5^2 + 12^2 = 13^2$.B. The Triangle Is A Right

Introduction

In geometry, a right triangle is a triangle with one angle that measures 90 degrees. This type of triangle has several unique properties, including the relationship between the lengths of its sides. In this article, we will explore how to determine whether a given triangle is a right triangle using the Pythagorean theorem.

The Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem states that in a right triangle with legs of length $a$ and $b$, and a hypotenuse of length $c$, the following equation holds:

$$a^2 + b^2 = c^2$$

This equation can be used to determine whether a given triangle is a right triangle by checking if the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side.

Example: A Triangle with Side Lengths 5 cm, 13 cm, and 12 cm

Let's consider a triangle with side lengths $5 \, \text{cm}, 13 \, \text{cm}$, and $12 \, \text{cm}$. We want to determine whether this triangle is a right triangle using the Pythagorean theorem.

Step 1: Identify the Lengths of the Sides

The lengths of the sides of the triangle are given as $5 \, \text{cm}, 13 \, \text{cm}$, and $12 \, \text{cm}$. We need to identify which side is the longest and which sides are the legs.

Step 2: Apply the Pythagorean Theorem

We will apply the Pythagorean theorem to the triangle by checking if the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side.

$$5^2 + 12^2 = 13^2$$

Step 3: Calculate the Squares of the Lengths

We need to calculate the squares of the lengths of the sides.

$$5^2 = 25$$

$$12^2 = 144$$

$$13^2 = 169$$

Step 4: Check if the Equation Holds

We will check if the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side.

$$25 + 144 = 169$$

Since the equation holds, we can conclude that the triangle is a right triangle.

Conclusion

In this article, we have explored how to determine whether a given triangle is a right triangle using the Pythagorean theorem. We have applied the theorem to a triangle with side lengths $5 \, \text{cm}, 13 \, \text{cm}$, and $12 \, \text{cm}$ and found that it is indeed a right triangle. The Pythagorean theorem is a powerful tool for determining the properties of triangles, and it has numerous applications in geometry and other fields.

Common Misconceptions

There are several common misconceptions about the Pythagorean theorem that can lead to incorrect conclusions. Here are a few examples:

• The theorem only applies to right triangles: This is not true. The Pythagorean theorem can be used to determine the properties of any triangle, not just right triangles.
• The theorem only applies to triangles with integer side lengths: This is not true. The Pythagorean theorem can be used to determine the properties of triangles with any side lengths, including non-integer values.
• The theorem is only used in geometry: This is not true. The Pythagorean theorem has numerous applications in other fields, including physics, engineering, and computer science.

Real-World Applications

The Pythagorean theorem has numerous real-world applications, including:

• Building design: The theorem is used to determine the properties of triangles in building design, including the height and length of walls and roofs.
• Physics: The theorem is used to determine the properties of triangles in physics, including the motion of objects and the behavior of waves.
• Computer science: The theorem is used to determine the properties of triangles in computer science, including the design of algorithms and data structures.

Conclusion

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem states that in a right triangle with legs of length $a$ and $b$, and a hypotenuse of length $c$, the following equation holds:

$$a^2 + b^2 = c^2$$

Q: What is a right triangle?

A: A right triangle is a triangle with one angle that measures 90 degrees. This type of triangle has several unique properties, including the relationship between the lengths of its sides.

Q: How do I use the Pythagorean theorem to determine if a triangle is a right triangle?

A: To use the Pythagorean theorem to determine if a triangle is a right triangle, you need to follow these steps:

1. Identify the lengths of the sides of the triangle.
2. Determine which side is the longest (the hypotenuse).
3. Check if the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side.

Q: What are the limitations of the Pythagorean theorem?

A: The Pythagorean theorem only applies to right triangles, and it only works for triangles with side lengths that can be expressed as integers or rational numbers. Additionally, the theorem does not provide information about the angles of the triangle.

Q: Can I use the Pythagorean theorem to determine the length of a side of a triangle?

A: Yes, you can use the Pythagorean theorem to determine the length of a side of a triangle. If you know the lengths of the other two sides, you can use the theorem to calculate the length of the third side.

Q: How do I apply the Pythagorean theorem to a triangle with non-integer side lengths?

A: To apply the Pythagorean theorem to a triangle with non-integer side lengths, you need to use the theorem in its general form:

$$a^2 + b^2 = c^2$$

This form of the theorem works for any triangle, regardless of the lengths of its sides.

Q: Can I use the Pythagorean theorem to determine the properties of a triangle in three dimensions?

A: Yes, you can use the Pythagorean theorem to determine the properties of a triangle in three dimensions. The theorem can be extended to three dimensions using the concept of vectors.

Q: What are some real-world applications of the Pythagorean theorem?

A: The Pythagorean theorem has numerous real-world applications, including:

• Building design: The theorem is used to determine the properties of triangles in building design, including the height and length of walls and roofs.
• Physics: The theorem is used to determine the properties of triangles in physics, including the motion of objects and the behavior of waves.
• Computer science: The theorem is used to determine the properties of triangles in computer science, including the design of algorithms and data structures.

Q: How do I remember the Pythagorean theorem?

A: There are several ways to remember the Pythagorean theorem, including:

• Using a mnemonic device: Create a mnemonic device, such as "A squared plus B squared equals C squared," to help you remember the theorem.
• Visualizing the theorem: Visualize the theorem by drawing a right triangle and labeling the sides.
• Practicing problems: Practice problems that involve the Pythagorean theorem to help you remember the theorem.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem has numerous applications in geometry and other fields, and it is a powerful tool for determining the properties of triangles. By understanding the theorem and its applications, we can gain a deeper appreciation for the beauty and complexity of mathematics.