Question

Explain how you can use the Pythagorean theorem to find the distance between any two points in a coordinate plane.

Short answer

The Pythagorean theorem can be used to find the distance between any two points in a coordinate plane by visualizing the two points as vertices of a right triangle. Then, calculate the length of the triangle's hypotenuse (which represents the direct distance between the two points) as: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step-by-step solution

Understand the Pythagorean Theorem

The Pythagorean theorem 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. This can be written as \(a^2 + b^2 = c^2\), where \(c\) represents the length of the hypotenuse, while \(a\) and \(b\) represent the lengths of the other two sides.

Coordinate Geometry Perspective

In coordinate geometry, any two distinct points (x_1, y_1) and (x_2, y_2) can be seen as vertices of a right triangle, where the hypotenuse of the triangle represents the distance between the two points. The other two sides of the triangle are parallel to the axes, and their lengths correspond to the absolute differences in the x-coordinates and the y-coordinates of these two points.

Use the Pythagorean Theorem

Position the two points as corners of a right triangle. The horizontal leg of the triangle would be the absolute difference of the x-coordinates of the two points, i.e., |x_2 - x_1|, and the vertical leg would be the absolute difference of the y-coordinates, i.e., |y_2 - y_1|. So, the length of the hypotenuse (which is the direct distance between the two points) can be calculated using the Pythagorean theorem: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Key concepts

Coordinate Geometry

Understanding the realm of coordinate geometry is crucial for visualizing and solving many mathematical and real-world problems. It allows us to represent geometric figures and the relationships between their points using a set of coordinates. This system uses a grid made up of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Each point on this plane can be described by an ordered pair (x, y), representing its horizontal and vertical positions respectively.

In the context of the Pythagorean theorem, cooridinate geometry empowers us to calculate the distance between any two points by transforming the problem into a question about right triangles. This capability is incredibly useful in a plethora of fields, from physics to computer graphics to navigation, and it all stems from viewing the space around us through this structured mathematical lens.

Hypotenuse Calculation

Hypotenuse calculation is a fundamental aspect of working with right triangles. In the context of the Pythagorean theorem, the hypotenuse is the longest side, opposite the right angle, and is key when it comes to determining the distance between two points on a grid. When two sides of a right triangle are known, we can easily calculate the length of the hypotenuse using the formula c = \( \sqrt{a^2 + b^2} \) . This concept isn't just theoretical; it's practical as well. For example, architects use hypotenuse calculations to determine the length of a ladder needed or to ensure the accuracy of their designs.

Distance Formula

The distance formula is an algebraic expression derived from the Pythagorean theorem and is used to determine the straight-line distance between two points on a coordinate plane. Given as d = \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) , the distance formula encapsulates the process of constructing a right triangle from the two points and calculating the length of the hypotenuse. More practically, this formula is indispensable in navigation (finding the shortest path between two locations) and in technology (ranging from plotting points on a map to developing video games). By understanding and applying this formula, students can easily move from the abstract world of theoretical geometry to practical applications in their everyday lives.