Reynaldo Rode His Bike 2 Miles North And 3 Miles East. Which Equation Should He Use To Find The Distance, D D D , That Takes Him Directly Back Home?A. 2 2 + 3 2 = D 2 2^2 + 3^2 = D^2 2 2 + 3 2 = D 2 B. 3 2 − 2 2 = Σ 2 3^2 - 2^2 = \sigma^2 3 2 − 2 2 = Σ 2 C. D 2 + 2 2 = 3 2 D^2 + 2^2 = 3^2 D 2 + 2 2 = 3 2 D.

Introduction

Reynaldo embarked on a bike ride, covering a distance of 2 miles north and 3 miles east. Now, he wants to find the shortest distance, $d$, that will take him directly back home. This problem can be approached using the Pythagorean theorem, which is a fundamental concept in geometry and trigonometry. In this article, we will explore the correct equation that Reynaldo should use to find the distance, $d$, that takes him directly back home.

Understanding 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. Mathematically, this can be expressed as:

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

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

Applying the Pythagorean Theorem to Reynaldo's Bike Ride

In Reynaldo's case, the two sides of the right-angled triangle are the 2 miles he rode north and the 3 miles he rode east. The distance, $d$, that he needs to travel to get back home is the length of the hypotenuse. Using the Pythagorean theorem, we can write the equation as:

$$2^2 + 3^2 = d^2$$

This equation represents the sum of the squares of the two sides (2 miles and 3 miles) equaling the square of the length of the hypotenuse (the distance, $d$, that Reynaldo needs to travel to get back home).

Evaluating the Answer Choices

Now, let's evaluate the answer choices to determine which one is correct.

• A. $2^2 + 3^2 = d^2$: This is the correct equation, as we derived it using the Pythagorean theorem.
• B. $3^2 - 2^2 = \sigma^2$: This equation is incorrect because it represents the difference of the squares of the two sides, not the sum.
• C. $d^2 + 2^2 = 3^2$: This equation is also incorrect because it represents the sum of the square of the hypotenuse and one of the sides, not the sum of the squares of the two sides.
• D.: This option is not provided.

Conclusion

In conclusion, Reynaldo should use the equation $2^2 + 3^2 = d^2$ to find the distance, $d$, that takes him directly back home. This equation is based on the Pythagorean theorem, which is a fundamental concept in geometry and trigonometry. By applying this theorem, we can solve problems involving right-angled triangles and find the length of the hypotenuse.

Real-World Applications

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

• Architecture: Architects use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in designing buildings and structures.
• Engineering: Engineers use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in designing bridges, roads, and other infrastructure projects.
• Navigation: Pilots and sailors use the Pythagorean theorem to calculate the distance between two points on a map, which is essential in navigation.

Final Thoughts

Q&A: Frequently Asked Questions About Reynaldo's Bike Ride

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry and trigonometry that 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 find the distance between two points?

A: To apply the Pythagorean theorem, you need to identify the two sides of the right-angled triangle and the length of the hypotenuse. Then, you can use the formula:

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

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

Q: What if I have a right-angled triangle with two sides of length 5 and 12? How do I find the length of the hypotenuse?

A: To find the length of the hypotenuse, you can use the Pythagorean theorem:

$$5^2 + 12^2 = c^2$$

$$25 + 144 = c^2$$

$$169 = c^2$$

$$c = \sqrt{169}$$

$$c = 13$$

So, the length of the hypotenuse is 13.

Q: Can I use the Pythagorean theorem to find the distance between two points on a map?

A: Yes, you can use the Pythagorean theorem to find the distance between two points on a map. For example, if you have a map with two points labeled A and B, and you know the distance between A and C and the distance between C and B, you can use the Pythagorean theorem to find the distance between A and B.

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

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

• Architecture: Architects use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in designing buildings and structures.
• Engineering: Engineers use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in designing bridges, roads, and other infrastructure projects.
• Navigation: Pilots and sailors use the Pythagorean theorem to calculate the distance between two points on a map, which is essential in navigation.

Q: Can I use the Pythagorean theorem to find the distance between two points in three dimensions?

A: Yes, you can use the Pythagorean theorem to find the distance between two points in three dimensions. However, you need to use the formula:

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

where $a$, $b$, and $c$ are the lengths of the three sides of the right-angled triangle, and $d$ is the length of the hypotenuse.

Q: What if I have a right-angled triangle with two sides of length 5 and 12, and the third side is 15? How do I find the length of the hypotenuse?

A: To find the length of the hypotenuse, you can use the Pythagorean theorem:

$$5^2 + 12^2 + 15^2 = d^2$$

$$25 + 144 + 225 = d^2$$

$$394 = d^2$$

$$d = \sqrt{394}$$

So, the length of the hypotenuse is approximately 19.8.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry and trigonometry that has numerous real-world applications. By understanding and applying this theorem, we can solve problems involving right-angled triangles and find the length of the hypotenuse.