Find The Distance Between The Points { (-47, 79)$}$ And { (-42, 67)$}$.Distance = { \square$}$

by ADMIN 95 views

===========================================================

Introduction

In mathematics, the distance between two points in a coordinate plane is a fundamental concept that has numerous applications in various fields, including geometry, trigonometry, and physics. The distance between two points can be calculated using the distance formula, which is derived from the Pythagorean theorem. In this article, we will explore how to find the distance between two points in a coordinate plane using the distance formula.

What is the Distance Formula?

The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points, and \sqrt{} represents the square root.

How to Use the Distance Formula

To use the distance formula, you need to follow these steps:

  1. Identify the coordinates of the two points: The first step is to identify the coordinates of the two points for which you want to find the distance. In this case, the coordinates of the two points are (−47,79)(-47, 79) and (−42,67)(-42, 67).
  2. Plug in the coordinates into the distance formula: Once you have identified the coordinates of the two points, you can plug them into the distance formula. In this case, the distance formula becomes:

d=((−42)−(−47))2+(67−79)2d = \sqrt{((-42) - (-47))^2 + (67 - 79)^2}

  1. Simplify the expression: After plugging in the coordinates, you need to simplify the expression by evaluating the expressions inside the parentheses and the square root.

Calculating the Distance

Now that we have the distance formula, let's calculate the distance between the two points.

d=((−42)−(−47))2+(67−79)2d = \sqrt{((-42) - (-47))^2 + (67 - 79)^2}

d=(5)2+(−12)2d = \sqrt{(5)^2 + (-12)^2}

d=25+144d = \sqrt{25 + 144}

d=169d = \sqrt{169}

d=13d = 13

Therefore, the distance between the points (−47,79)(-47, 79) and (−42,67)(-42, 67) is 1313 units.

Real-World Applications

The distance formula has numerous real-world applications, including:

  • Navigation: The distance formula is used in navigation to calculate the distance between two points on a map.
  • Physics: The distance formula is used in physics to calculate the distance traveled by an object.
  • Engineering: The distance formula is used in engineering to calculate the distance between two points in a design.

Conclusion

In conclusion, the distance formula is a powerful tool that can be used to calculate the distance between two points in a coordinate plane. By following the steps outlined in this article, you can use the distance formula to calculate the distance between two points. The distance formula has numerous real-world applications, including navigation, physics, and engineering.

Frequently Asked Questions

Q: What is the distance formula?

A: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane.

Q: How do I use the distance formula?

A: To use the distance formula, you need to identify the coordinates of the two points, plug them into the distance formula, and simplify the expression.

Q: What are the real-world applications of the distance formula?

A: The distance formula has numerous real-world applications, including navigation, physics, and engineering.

Q: How do I calculate the distance between two points?

A: To calculate the distance between two points, you need to use the distance formula and follow the steps outlined in this article.

Glossary

  • Coordinate plane: A two-dimensional plane that is used to represent points and lines.
  • Distance formula: A mathematical formula that calculates the distance between two points in a coordinate plane.
  • Pythagorean theorem: A mathematical theorem that states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

References

  • "Mathematics for Dummies" by Mark Ryan
  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Trigonometry: A First Course" by Michael Corral

===========================================================

Introduction

The distance formula is a fundamental concept in mathematics that has numerous applications in various fields, including geometry, trigonometry, and physics. In our previous article, we explored how to find the distance between two points in a coordinate plane using the distance formula. In this article, we will answer some of the most frequently asked questions about the distance formula.

Q&A

Q: What is the distance formula?

A: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where dd is the distance between the two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points, and \sqrt{} represents the square root.

Q: How do I use the distance formula?

A: To use the distance formula, you need to follow these steps:

  1. Identify the coordinates of the two points: The first step is to identify the coordinates of the two points for which you want to find the distance.
  2. Plug in the coordinates into the distance formula: Once you have identified the coordinates of the two points, you can plug them into the distance formula.
  3. Simplify the expression: After plugging in the coordinates, you need to simplify the expression by evaluating the expressions inside the parentheses and the square root.

Q: What are the real-world applications of the distance formula?

A: The distance formula has numerous real-world applications, including:

  • Navigation: The distance formula is used in navigation to calculate the distance between two points on a map.
  • Physics: The distance formula is used in physics to calculate the distance traveled by an object.
  • Engineering: The distance formula is used in engineering to calculate the distance between two points in a design.

Q: How do I calculate the distance between two points?

A: To calculate the distance between two points, you need to use the distance formula and follow the steps outlined above.

Q: What is the difference between the distance formula and the Pythagorean theorem?

A: The distance formula and the Pythagorean theorem are related but distinct concepts. The Pythagorean theorem is a mathematical theorem that states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane.

Q: Can I use the distance formula to calculate the distance between three points?

A: No, the distance formula is used to calculate the distance between two points in a coordinate plane. If you want to calculate the distance between three points, you need to use a different formula or method.

Q: How do I handle negative coordinates in the distance formula?

A: When working with negative coordinates, you need to be careful when plugging them into the distance formula. Make sure to follow the order of operations and simplify the expression carefully.

Q: Can I use the distance formula to calculate the distance between two points in three-dimensional space?

A: No, the distance formula is used to calculate the distance between two points in a coordinate plane. If you want to calculate the distance between two points in three-dimensional space, you need to use a different formula or method.

Conclusion

In conclusion, the distance formula is a powerful tool that can be used to calculate the distance between two points in a coordinate plane. By following the steps outlined in this article, you can use the distance formula to answer a variety of questions and solve problems. Remember to be careful when working with negative coordinates and to use the correct formula or method for calculating distances in three-dimensional space.

Glossary

  • Coordinate plane: A two-dimensional plane that is used to represent points and lines.
  • Distance formula: A mathematical formula that calculates the distance between two points in a coordinate plane.
  • Pythagorean theorem: A mathematical theorem that states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

References

  • "Mathematics for Dummies" by Mark Ryan
  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Trigonometry: A First Course" by Michael Corral