Find The Distance Between The Points { (3,4)$}$ And { (-1,0)$}$.A. 8 B. ${ 2 \sqrt{5}\$} C. 4 D. ${ 4 \sqrt{2}\$}

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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 Pythagorean theorem, which states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides. In this article, we will explore how to find the distance between two points in a coordinate plane using the Pythagorean theorem.

The Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in mathematics that has been used for centuries to calculate the distance between two points in a coordinate plane. The theorem states that the square of the length of the hypotenuse of a right-angled triangle 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 of the triangle, and c is the length of the hypotenuse.

Calculating the Distance Between Two Points

To calculate the distance between two points in a coordinate plane, we can use the Pythagorean theorem. Let's consider two points, (x1, y1) and (x2, y2), in a coordinate plane. The distance between these two points can be calculated using the following formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Example

Let's consider two points, (3, 4) and (-1, 0), in a coordinate plane. We can use the formula above to calculate the distance between these two points.

d = √((-1 - 3)^2 + (0 - 4)^2) d = √((-4)^2 + (-4)^2) d = √(16 + 16) d = √32 d = √(16 * 2) d = 4√2

Conclusion

In conclusion, the distance between two points in a coordinate plane can be calculated using the Pythagorean theorem. The formula for calculating the distance between two points is d = √((x2 - x1)^2 + (y2 - y1)^2), where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points. In the example above, we calculated the distance between the points (3, 4) and (-1, 0) using this formula, and found that the distance is 4√2.

Answer

The correct answer is D. [4√2].

Discussion

  • What is the Pythagorean theorem, and how is it used to calculate the distance between two points in a coordinate plane?
  • How do you calculate the distance between two points in a coordinate plane using the Pythagorean theorem?
  • What is the formula for calculating the distance between two points in a coordinate plane?
  • How do you apply the formula to calculate the distance between two points in a coordinate plane?

Related Topics

  • Coordinate geometry
  • Trigonometry
  • Pythagorean theorem
  • Distance formula
  • Coordinate plane

References

Frequently Asked Questions

Q: What is the Pythagorean theorem, and how is it used to calculate the distance between two points in a coordinate plane?

A: The Pythagorean theorem is a fundamental concept in mathematics that states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides. It is used to calculate the distance between two points in a coordinate plane by finding the length of the hypotenuse of a right-angled triangle formed by the two points.

Q: How do you calculate the distance between two points in a coordinate plane using the Pythagorean theorem?

A: To calculate the distance between two points in a coordinate plane using the Pythagorean theorem, you need to find the difference in the x-coordinates and the difference in the y-coordinates of the two points. Then, you square these differences and add them together. The square root of the sum is the distance between the two points.

Q: What is the formula for calculating the distance between two points in a coordinate plane?

A: The formula for calculating the distance between two points in a coordinate plane is:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: How do you apply the formula to calculate the distance between two points in a coordinate plane?

A: To apply the formula, you need to substitute the coordinates of the two points into the formula and simplify. For example, if you want to find the distance between the points (3, 4) and (-1, 0), you would substitute x1 = 3, y1 = 4, x2 = -1, and y2 = 0 into the formula.

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 general principle that applies to all right-angled triangles, while the distance formula is a specific application of the Pythagorean theorem to calculate the distance between two points in a coordinate plane.

Q: Can you give an example of how to use the distance formula to calculate the distance between two points?

A: Yes, let's consider the points (3, 4) and (-1, 0). To find the distance between these points, we would substitute x1 = 3, y1 = 4, x2 = -1, and y2 = 0 into the formula:

d = √((-1 - 3)^2 + (0 - 4)^2) d = √((-4)^2 + (-4)^2) d = √(16 + 16) d = √32 d = √(16 * 2) d = 4√2

Q: What is the significance of the distance formula in real-world applications?

A: The distance formula has numerous real-world applications, including navigation, geography, and physics. It is used to calculate distances between locations, determine the length of routes, and model the behavior of objects in motion.

Q: Can you provide some tips for remembering the distance formula?

A: Yes, here are some tips for remembering the distance formula:

  • Write the formula down and practice using it with different examples.
  • Visualize the formula as a geometric shape, such as a right-angled triangle.
  • Break down the formula into smaller parts and focus on one part at a time.
  • Use mnemonic devices, such as "d equals the square root of the difference in x-coordinates squared plus the difference in y-coordinates squared."

Q: What are some common mistakes to avoid when using the distance formula?

A: Some common mistakes to avoid when using the distance formula include:

  • Forgetting to square the differences in x-coordinates and y-coordinates.
  • Failing to add the squared differences together.
  • Not taking the square root of the sum.
  • Making errors when substituting the coordinates of the two points into the formula.

Q: Can you provide some resources for further learning on the distance formula?

A: Yes, here are some resources for further learning on the distance formula:

  • Khan Academy: Distance formula
  • Math Open Reference: Distance formula
  • Purplemath: Distance formula
  • Wolfram MathWorld: Distance formula

Conclusion

In conclusion, the distance formula is a powerful tool for calculating distances between points in a coordinate plane. By understanding the Pythagorean theorem and the distance formula, you can apply these concepts to a wide range of real-world problems. Remember to practice using the formula with different examples, visualize the formula as a geometric shape, and break down the formula into smaller parts to make it easier to remember.