Calculating Distance Using the Pythagorean Theorem

Oct 13, 2025 by miodeya.com 22 views

Introduction

In mathematics, particularly in geometry and trigonometry, calculating the distance between two points in a coordinate plane is a fundamental concept. This article will explore the different methods of calculating the distance between two points and provide a step-by-step guide on how to choose the correct expression.

What is Distance?

Distance is a measure of the length between two points in a coordinate plane. It is an essential concept in mathematics, as it helps us understand the relationship between different points and shapes. In a coordinate plane, distance is typically measured using the Pythagorean theorem, which states that 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.

Calculating Distance Using the Pythagorean Theorem

The Pythagorean theorem is a powerful tool for calculating distance between two points in a coordinate plane. The formula for the Pythagorean theorem is:

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.

To calculate the distance between two points (x1, y1) and (x2, y2), we can use the following formula:

distance = √((x2 - x1)^2 + (y2 - y1)^2)

Choosing the Correct Expression

Now that we have discussed the concept of distance and the Pythagorean theorem, let's examine the different expressions given in the options.

Option A: (4-7)^2 + (6-3)^2

This expression is incorrect because it does not use the correct formula for calculating distance. The correct formula is √((x2 - x1)^2 + (y2 - y1)^2), not (x2 - x1)^2 + (y2 - y1)^2.

Option B: √((4-7)^2 + (6+3)^2)

This expression is also incorrect because it uses the wrong formula and also has a mistake in the addition of y-coordinates. The correct formula is √((x2 - x1)^2 + (y2 - y1)^2), not √((x2 - x1)^2 + (y2 + y1)^2).

Option C: (4-7)^2 + (6+3)^2

This expression is incorrect because it does not use the correct formula for calculating distance. The correct formula is √((x2 - x1)^2 + (y2 - y1)^2), not (x2 - x1)^2 + (y2 + y1)^2.

Option D: √((4-7)^2 + (6-3)^2)

This expression is the correct formula for calculating distance between two points in a coordinate plane. It uses the Pythagorean theorem and the correct formula for calculating distance.

Conclusion

Calculating distance between two points in a coordinate plane is a fundamental concept in mathematics. By using the Pythagorean theorem and the correct formula, we can calculate the distance between two points. In this article, we have discussed the concept of distance, the Pythagorean theorem, and how to choose the correct expression for calculating distance. We have also examined the different options and identified the correct expression.

Final Answer

The final answer is:

Option D: √((4-7)^2 + (6-3)^2)

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

A: The formula for calculating distance between two points (x1, y1) and (x2, y2) is:

distance = √((x2 - x1)^2 + (y2 - y1)^2)

Q: What is the Pythagorean theorem, and how is it used in calculating distance?

A: The Pythagorean theorem is a mathematical concept that states that 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. In the context of calculating distance, the Pythagorean theorem is used to find the length of the hypotenuse, which is the distance between the two points.

Q: What is the difference between the correct and incorrect expressions for calculating distance?

A: The correct expression for calculating distance is:

√((x2 - x1)^2 + (y2 - y1)^2)

The incorrect expressions are:

(x2 - x1)^2 + (y2 - y1)^2
√((x2 - x1)^2 + (y2 + y1)^2)
(x2 - x1)^2 + (y2 + y1)^2

The main difference between the correct and incorrect expressions is the use of the square root symbol (√) and the correct formula for calculating distance.

Q: How do I choose the correct expression for calculating distance?

A: To choose the correct expression for calculating distance, follow these steps:

Identify the coordinates of the two points (x1, y1) and (x2, y2).
Use the correct formula for calculating distance: √((x2 - x1)^2 + (y2 - y1)^2).
Plug in the values of the coordinates into the formula.
Simplify the expression to find the distance between the two points.

Q: What are some common mistakes to avoid when calculating distance?

A: Some common mistakes to avoid when calculating distance include:

Using the wrong formula for calculating distance.
Not using the correct coordinates of the two points.
Not simplifying the expression to find the distance.
Not using the square root symbol (√) when necessary.

Q: Can I use other methods to calculate distance, such as using a calculator or a computer program?

A: Yes, you can use other methods to calculate distance, such as using a calculator or a computer program. However, it's essential to understand the underlying mathematics and the correct formula for calculating distance to ensure accuracy and precision.

Q: How do I apply the concept of distance to real-world problems?

A: The concept of distance can be applied to various real-world problems, such as:

Calculating the distance between two cities or locations.
Finding the shortest path between two points.
Determining the distance between two objects or points in a coordinate plane.
Calculating the distance between two points in a three-dimensional space.

By understanding the concept of distance and how to calculate it, you can apply this knowledge to a wide range of real-world problems and scenarios.