In The Xy-plane, The Distance Between Points A ( 1 , 2 2 A(1,2 \sqrt{2} A ( 1 , 2 2 ​ ] And B\left(\frac{\sqrt{3}}{2}, 6\right ] Is Approximately Equal To What?(A) 3.2 (B) 3.7 (C) 5.6 (D) 8.8 (E) 9.0

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Introduction

In mathematics, the distance between two points in a plane is a fundamental concept 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 calculate the distance between two points in the xy-plane and apply this concept to a specific problem.

The Distance Formula

The distance formula is used to find the distance between two points (x1, y1) and (x2, y2) in a plane. The formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where d is the distance between the two points.

Calculating Distance Between Points A and B

We are given two points A(1, 2√2) and B(√3/2, 6) in the xy-plane. To find the distance between these two points, we will use the distance formula.

First, we need to substitute the coordinates of points A and B into the distance formula:

d = √((√3/2 - 1)^2 + (6 - 2√2)^2)

To simplify the calculation, we will expand the squares inside the parentheses:

d = √((√3/2 - 1)(√3/2 - 1) + (6 - 2√2)(6 - 2√2))

Expanding the products inside the parentheses:

d = √((3/4 - √3/2 + 1) + (36 - 24√2 + 8))

Combine like terms:

d = √((3/4 - √3/2 + 1) + (44 - 24√2))

Now, we need to simplify the expression inside the square root:

d = √((3 - 2√3 + 4) + (44 - 24√2))

Combine like terms:

d = √(7 - 2√3 + 44 - 24√2)

d = √(51 - 2√3 - 24√2)

To simplify the expression further, we will use the fact that √3 ≈ 1.732 and √2 ≈ 1.414:

d ≈ √(51 - 2(1.732) - 24(1.414))

d ≈ √(51 - 3.464 - 33.936)

d ≈ √(13.6)

d ≈ 3.68

Conclusion

In this article, we discussed how to calculate the distance between two points in the xy-plane using the distance formula. We applied this concept to a specific problem and calculated the distance between points A(1, 2√2) and B(√3/2, 6). The approximate distance between these two points is 3.68.

Answer

The approximate distance between points A and B is 3.7.

Discussion

This problem is a classic example of how to apply the distance formula in a real-world scenario. The distance formula is a fundamental concept in mathematics that has numerous applications in various fields such as geometry, trigonometry, and physics. In this problem, we used the distance formula to calculate the distance between two points in the xy-plane. The result is an approximate value of 3.7, which is one of the answer choices.

Additional Resources

For more information on the distance formula and its applications, please refer to the following resources:

References

Introduction

In our previous article, we discussed how to calculate the distance between two points in the xy-plane using the distance formula. In this article, we will provide a Q&A section to help you better understand the concept and apply it to real-world scenarios.

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to find the distance between two points (x1, y1) and (x2, y2) in a plane. The formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Q: How do I apply the distance formula?

A: To apply the distance formula, you need to substitute the coordinates of the two points into the formula and simplify the expression inside the square root.

Q: What if the coordinates are in decimal form?

A: If the coordinates are in decimal form, you can simply substitute them into the distance formula and simplify the expression inside the square root.

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

A: No, the distance formula is only applicable to points in a 2D plane. To find the distance between two points in 3D space, you need to use the 3D distance formula, which is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Q: How do I calculate the distance between two points on a coordinate grid?

A: To calculate the distance between two points on a coordinate grid, you can use the distance formula. However, you can also use the Pythagorean theorem to find the distance between the two points.

Q: Can I use the distance formula to find the distance between two points on a non-coordinate grid?

A: No, the distance formula is only applicable to points on a coordinate grid. To find the distance between two points on a non-coordinate grid, you need to use a different method, such as the Pythagorean theorem or a ruler.

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

A: To calculate the distance between two points with negative coordinates, you can use the distance formula. However, you need to be careful when simplifying the expression inside the square root.

Q: Can I use the distance formula to find the distance between two points with decimal coordinates?

A: Yes, you can use the distance formula to find the distance between two points with decimal coordinates. Simply substitute the decimal coordinates into the formula and simplify the expression inside the square root.

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

A: To calculate the distance between two points on a graph, you can use the distance formula. However, you need to be careful when identifying the coordinates of the two points.

Q: Can I use the distance formula to find the distance between two points on a non-linear graph?

A: No, the distance formula is only applicable to points on a linear graph. To find the distance between two points on a non-linear graph, you need to use a different method, such as the Pythagorean theorem or a ruler.

Conclusion

In this Q&A article, we provided answers to common questions about calculating the distance between two points in the xy-plane using the distance formula. We hope this article has helped you better understand the concept and apply it to real-world scenarios.

Additional Resources

For more information on the distance formula and its applications, please refer to the following resources:

References