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

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 whether a triangle with side lengths $5 \text{ cm}$, $13 \text{ cm}$, and $12 \text{ cm}$ is a right triangle.

Understanding the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that helps determine whether a triangle is a right triangle or not. The 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. Mathematically, this can be expressed as:

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

where $a$ and $b$ are the lengths of the two sides that form the right angle, and $c$ is the length of the hypotenuse.

Applying the Pythagorean Theorem to the Given Triangle

Now, let's apply the Pythagorean theorem to the given triangle with side lengths $5 \text{ cm}$, $13 \text{ cm}$, and $12 \text{ cm}$. We can use the theorem to check if the triangle is a right triangle.

First, we need to identify the sides of the triangle. Let's assume that the side with length $5 \text{ cm}$ is one of the sides that form the right angle, and the side with length $12 \text{ cm}$ is the other side. The side with length $13 \text{ cm}$ is the hypotenuse.

Using the Pythagorean theorem, we can calculate the sum of the squares of the lengths of the two sides that form the right angle:

$$5^2 + 12^2 = 25 + 144 = 169$$

Now, let's calculate the square of the length of the hypotenuse:

$$13^2 = 169$$

As we can see, the sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse. This suggests that the triangle is a right triangle.

Conclusion

In conclusion, the triangle with side lengths $5 \text{ cm}$, $13 \text{ cm}$, and $12 \text{ cm}$ is a right triangle because it satisfies the Pythagorean theorem. The sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse.

The Correct Answer

The correct answer is:

A. The triangle is a right triangle because $5^2 + 12^2 = 13^2$.

Why the Other Option is Incorrect

The other option is incorrect because it states that the triangle is not a right triangle. However, as we have shown, the triangle satisfies the Pythagorean theorem, which means that it is a right triangle.

Final Thoughts

In this article, we have explored whether a triangle with side lengths $5 \text{ cm}$, $13 \text{ cm}$, and $12 \text{ cm}$ is a right triangle. We have applied the Pythagorean theorem to check if the triangle is a right triangle, and we have found that it is indeed a right triangle. This demonstrates the importance of the Pythagorean theorem in geometry and its role in determining the nature of triangles.

References

• [1] "Pythagorean Theorem." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/topic/Pythagorean-theorem.
• [2] "Right Triangle." Math Open Reference, mathopenref.com/righttriangle.html.

Additional Resources

• [1] Khan Academy. "Pythagorean Theorem." Khan Academy, www.khanacademy.org/math/geometry/geometry-trigonometry/geometry-trigonometry/v/pythagorean-theorem.
• [2] Mathway. "Pythagorean Theorem Calculator." Mathway, mathway.com/calculator/pythagorean-theorem-calculator.
  Frequently Asked Questions: Determining the Nature of a Triangle ====================================================================

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that helps determine whether a triangle is a right triangle or not. The 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 I apply the Pythagorean theorem to a triangle?

A: To apply the Pythagorean theorem to a triangle, you need to identify the sides of the triangle and calculate the sum of the squares of the lengths of the two sides that form the right angle. Then, you need to calculate the square of the length of the hypotenuse. If the sum of the squares of the lengths of the two sides that form the right angle is equal to the square of the length of the hypotenuse, then the triangle is a right triangle.

Q: What are the three sides of a right triangle?

A: The three sides of a right triangle are:

• The two sides that form the right angle (these are called the legs of the triangle)
• The side opposite the right angle (this is called the hypotenuse)

Q: How do I determine which side is the hypotenuse?

A: The hypotenuse is the side opposite the right angle. To determine which side is the hypotenuse, you need to identify the right angle and then look for the side that is opposite it.

Q: Can a triangle have more than one right angle?

A: No, a triangle cannot have more than one right angle. A right triangle by definition has one right angle, and the other two angles are acute angles.

Q: Can a triangle have a right angle and still not be a right triangle?

A: No, a triangle cannot have a right angle and still not be a right triangle. If a triangle has a right angle, then it is a right triangle.

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

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

• Building design: Architects use the Pythagorean theorem to calculate the height of buildings and the length of shadows.
• Physics: The Pythagorean theorem is used to calculate the distance and speed of objects in motion.
• Engineering: The Pythagorean theorem is used to calculate the stress and strain on materials.

Q: Can I use the Pythagorean theorem to calculate the area of a triangle?

A: No, the Pythagorean theorem is used to calculate the length of the sides of a triangle, not the area. To calculate the area of a triangle, you need to use a different formula.

Q: Can I use the Pythagorean theorem to calculate the perimeter of a triangle?

A: No, the Pythagorean theorem is used to calculate the length of the sides of a triangle, not the perimeter. To calculate the perimeter of a triangle, you need to add up the lengths of all three sides.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that helps determine whether a triangle is a right triangle or not. By applying the theorem to a triangle, you can determine whether it is a right triangle or not. The Pythagorean theorem has many real-world applications, including building design, physics, and engineering.

References

• [1] "Pythagorean Theorem." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/topic/Pythagorean-theorem.
• [2] "Right Triangle." Math Open Reference, mathopenref.com/righttriangle.html.

Additional Resources

• [1] Khan Academy. "Pythagorean Theorem." Khan Academy, www.khanacademy.org/math/geometry/geometry-trigonometry/geometry-trigonometry/v/pythagorean-theorem.
• [2] Mathway. "Pythagorean Theorem Calculator." Mathway, mathway.com/calculator/pythagorean-theorem-calculator.