Calculate Distance With Formula D=√(x-a)²+(y-b)²

Jul 11, 2025 by ADMIN 49 views

Calculating Distance Using the Distance Formula

In mathematics, especially in coordinate geometry, the distance formula is a crucial tool for determining the distance between two points in a coordinate plane. This formula is derived from the Pythagorean theorem and is widely used in various fields, including navigation, engineering, and computer graphics. This article will walk you through a step-by-step process of using the distance formula to calculate the distance between two points. We will apply the formula to a specific problem, ensuring a clear understanding of its application. This comprehensive guide aims to make the concept accessible and easy to grasp, even for those who are new to coordinate geometry. Understanding the distance formula is not just about plugging in numbers; it's about understanding the underlying geometric principles and how they translate into a practical mathematical tool. Mastering this formula opens doors to more advanced topics in geometry and beyond.

The distance formula is a direct application of the Pythagorean theorem, which relates the sides of a right triangle. In a coordinate plane, the distance between two points can be visualized as the hypotenuse of a right triangle, where the legs are the differences in the x-coordinates and the y-coordinates. The formula is expressed as:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Where:

$d$ is the distance between the two points.
$(x_1, y_1)$ are the coordinates of the first point.
$(x_2, y_2)$ are the coordinates of the second point.

The distance formula calculates the distance by finding the square root of the sum of the squares of the differences in the x and y coordinates. This formula is derived from the Pythagorean theorem ( $a^2 + b^2 = c^2$ ), where 'a' and 'b' are the lengths of the legs of a right triangle, and 'c' is the length of the hypotenuse. In the context of coordinate geometry, the difference in x-coordinates ( $x_2 - x_1$ ) and the difference in y-coordinates ( $y_2 - y_1$ ) represent the lengths of the legs of the right triangle, and the distance 'd' between the two points represents the length of the hypotenuse. Understanding this connection to the Pythagorean theorem provides a deeper insight into why the distance formula works. It's not just a formula to memorize; it's a geometric principle expressed in algebraic terms. The distance formula is versatile and applicable in various contexts, from simple geometric problems to complex spatial calculations in fields like physics and engineering. Its foundation in the Pythagorean theorem makes it a fundamental concept in mathematics.

We are given the distance formula:

$d = \sqrt{(x - a)^2 + (y - b)^2}$

And the following values:

$a = -4$
$b = -10$
$x = 0$
$y = 9$

Our goal is to calculate the distance d using the provided formula and values. This problem is a straightforward application of the distance formula, but it's crucial to pay attention to the signs and order of operations to arrive at the correct answer. The given values represent the coordinates of two points in a plane, and the distance formula will allow us to find the length of the straight line connecting these points. This type of problem is fundamental in coordinate geometry and serves as a building block for more complex geometric calculations. By carefully substituting the given values into the formula and following the correct algebraic steps, we can determine the distance d accurately. This exercise reinforces the importance of precision in mathematical calculations and highlights the practical application of the distance formula in finding the spatial separation between two points.

Substitute the given values into the distance formula: $d = \sqrt{(0 - (-4))^2 + (9 - (-10))^2}$

This step involves replacing the variables in the formula with their corresponding numerical values. It's crucial to handle the negative signs carefully to avoid errors. The expression inside the square root represents the sum of the squares of the differences in the x and y coordinates. This substitution sets the stage for the subsequent arithmetic operations that will lead to the final answer. Accuracy in this step is paramount, as any mistake in substitution will propagate through the rest of the calculation. This step is a direct application of the formula and demonstrates the importance of understanding the notation and meaning of each variable. It's a fundamental step in problem-solving that translates the abstract formula into a concrete calculation.
Simplify the expressions inside the parentheses: $d = \sqrt{(0 + 4)^2 + (9 + 10)^2}$ $d = \sqrt{(4)^2 + (19)^2}$

This step simplifies the expressions within the parentheses by performing the subtractions. Pay close attention to the signs; subtracting a negative number is the same as adding its positive counterpart. This simplification makes the subsequent squaring operations easier to manage. The goal is to reduce the complexity of the expression inside the square root before proceeding with further calculations. This step highlights the importance of following the order of operations (PEMDAS/BODMAS), where operations inside parentheses are performed first. Accurate simplification in this step is crucial for maintaining the integrity of the calculation and avoiding errors in the final result.
Square the numbers: $d = \sqrt{16 + 361}$

In this step, we square the simplified values from the previous step. Squaring a number means multiplying it by itself. The squares of 4 and 19 are calculated to further simplify the expression under the square root. This step is a straightforward arithmetic operation but is crucial for progressing towards the final solution. The squared values represent the contributions of the differences in x and y coordinates to the overall distance. This step demonstrates the application of the exponentiation operation and its role in the distance formula. Accurate squaring is essential for maintaining the correctness of the calculation and arriving at the correct distance value.
Add the numbers under the square root: $d = \sqrt{377}$

This step involves adding the two numbers obtained after squaring. The sum under the square root represents the total squared distance between the two points. This addition combines the contributions from both the x and y coordinates into a single value. This step simplifies the expression further, bringing us closer to the final answer. Accurate addition is crucial for ensuring the correctness of the result, as it is the final operation before taking the square root. This step highlights the importance of basic arithmetic skills in mathematical problem-solving and their role in applying formulas effectively.
Calculate the square root: $d ≈ 19.42$

This final step involves finding the square root of the sum calculated in the previous step. The square root of 377 is approximately 19.42 when rounded to two decimal places. This value represents the distance between the two points in the coordinate plane. The square root operation is the inverse of squaring and brings us back to the original unit of measurement for distance. This step completes the calculation and provides the final answer to the problem. The result is an approximation because the square root of 377 is an irrational number, meaning its decimal representation is non-terminating and non-repeating. Rounding to two decimal places provides a practical and usable value for the distance.

Therefore, the distance d between the points, rounded to two decimal places, is approximately 19.42.

This final answer represents the solution to the problem, providing the numerical value for the distance between the two points specified by the given coordinates. The rounding to two decimal places is a common practice to provide a practical and manageable value for the distance. This answer is the culmination of all the previous steps, each of which was essential for arriving at the correct solution. The process of calculating the distance involved substituting values into the formula, simplifying expressions, performing arithmetic operations, and finally, taking the square root. This comprehensive approach demonstrates the application of the distance formula and the importance of accurate calculations in mathematical problem-solving. The final answer provides a concrete measure of the spatial separation between the two points, which can be used in various applications, such as mapping, navigation, and geometry.

In this article, we successfully calculated the distance d between two points using the distance formula. We followed a step-by-step approach, starting with the substitution of given values, simplifying expressions, performing arithmetic operations, and finally, calculating the square root. The final answer, rounded to two decimal places, is approximately 19.42. This exercise demonstrates the practical application of the distance formula and highlights the importance of accurate calculations in mathematics. The distance formula is a fundamental tool in coordinate geometry and has wide-ranging applications in various fields. Understanding and applying this formula effectively is crucial for solving geometric problems and for more advanced mathematical concepts. The ability to calculate the distance between two points is not only a valuable skill in mathematics but also a useful tool in real-world applications, such as navigation, mapping, and spatial analysis. Mastering this formula provides a solid foundation for further exploration in geometry and related fields.