Ii) Find The Distance Between The Pair Of Points: { A(-8,1)$}$, { B(6,1)$}$

by ADMIN 76 views

Introduction

In mathematics, the distance between two points in a coordinate plane is a fundamental concept that is used in various fields such as 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 discuss how to find the distance between two points in a coordinate plane using the distance formula.

The Distance Formula

The distance formula is used to find 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.

Step-by-Step Solution

To find the distance between two points, we can follow these steps:

  1. Identify the coordinates of the two points: In this case, the coordinates of the two points are A(−8,1)A(-8, 1) and B(6,1)B(6, 1).
  2. Plug in the coordinates into the distance formula: We can plug in the coordinates of the two points into the distance formula to get:

d=(6−(−8))2+(1−1)2d = \sqrt{(6 - (-8))^2 + (1 - 1)^2}

  1. Simplify the expression: We can simplify the expression by evaluating the expressions inside the parentheses:

d=(14)2+(0)2d = \sqrt{(14)^2 + (0)^2}

  1. Evaluate the square root: We can evaluate the square root of the expression to get:

d=196d = \sqrt{196}

  1. Simplify the square root: We can simplify the square root by evaluating the square root of 196:

d=14d = 14

Conclusion

In this article, we discussed how to find the distance between two points in a coordinate plane using the distance formula. We followed a step-by-step approach to find the distance between the points A(−8,1)A(-8, 1) and B(6,1)B(6, 1). The distance between the two points is 14 units.

Example Problems

Here are some example problems that you can try to practice finding the distance between two points in a coordinate plane:

  • Find the distance between the points A(2,3)A(2, 3) and B(4,5)B(4, 5).
  • Find the distance between the points A(−2,1)A(-2, 1) and B(3,4)B(3, 4).
  • Find the distance between the points A(1,2)A(1, 2) and B(3,1)B(3, 1).

Tips and Tricks

Here are some tips and tricks that you can use to find the distance between two points in a coordinate plane:

  • Make sure to identify the coordinates of the two points correctly.
  • Plug in the coordinates into the distance formula correctly.
  • Simplify the expression inside the square root correctly.
  • Evaluate the square root correctly.

Common Mistakes

Here are some common mistakes that you can avoid when finding the distance between two points in a coordinate plane:

  • Not identifying the coordinates of the two points correctly.
  • Not plugging in the coordinates into the distance formula correctly.
  • Not simplifying the expression inside the square root correctly.
  • Not evaluating the square root correctly.

Real-World Applications

The distance formula has many real-world applications in fields such as:

  • Navigation: The distance formula is used in navigation to find the distance between two points on a map.
  • Surveying: The distance formula is used in surveying to find the distance between two points on a property.
  • Physics: The distance formula is used in physics to find the distance between two points in a coordinate plane.

Conclusion

Introduction

In our previous article, we discussed how to find the distance between two points in a coordinate plane using the distance formula. In this article, we will answer some frequently asked questions about finding the distance between two points in a coordinate plane.

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to find 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}

Q: How do I use the distance formula?

A: To use the distance formula, you need to identify the coordinates of the two points and plug them into the formula. Then, simplify the expression inside the square root and evaluate the square root to find the distance between the two points.

Q: What are the coordinates of the two points?

A: The coordinates of the two points are the x and y values that locate the points on the coordinate plane. For example, if the two points are A(2, 3) and B(4, 5), the coordinates of the two points are (2, 3) and (4, 5).

Q: How do I simplify the expression inside the square root?

A: To simplify the expression inside the square root, you need to evaluate the expressions inside the parentheses and then simplify the resulting expression. For example, if the expression inside the square root is (14)^2 + (0)^2, you can simplify it to 196 + 0 = 196.

Q: How do I evaluate the square root?

A: To evaluate the square root, you need to find the number that, when multiplied by itself, gives the value inside the square root. For example, if the value inside the square root is 196, you can evaluate the square root to find the number that, when multiplied by itself, gives 196.

Q: What are some common mistakes to avoid when finding the distance between two points?

A: Some common mistakes to avoid when finding the distance between two points include:

  • Not identifying the coordinates of the two points correctly
  • Not plugging in the coordinates into the distance formula correctly
  • Not simplifying the expression inside the square root correctly
  • Not evaluating the square root correctly

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

A: The distance formula has many real-world applications in fields such as:

  • Navigation: The distance formula is used in navigation to find the distance between two points on a map.
  • Surveying: The distance formula is used in surveying to find the distance between two points on a property.
  • Physics: The distance formula is used in physics to find the distance between two points in a coordinate plane.

Q: Can I use the distance formula to find the distance between two points in 3D space?

A: Yes, you can use the distance formula to find the distance between two points in 3D space. However, you need to use the 3D distance formula, which is given by:

d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Q: Can I use the distance formula to find the distance between two points on a circle or an ellipse?

A: Yes, you can use the distance formula to find the distance between two points on a circle or an ellipse. However, you need to use the formula for the distance between two points on a circle or an ellipse, which is given by:

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

where (x1, y1) and (x2, y2) are the coordinates of the two points on the circle or ellipse.

Conclusion

In conclusion, finding the distance between two points in a coordinate plane is a fundamental concept in mathematics that has many real-world applications. The distance formula is used to find the distance between two points, and it is derived from the Pythagorean theorem. By following a step-by-step approach and avoiding common mistakes, you can find the distance between two points in a coordinate plane with ease.