Name: finding distance on a coordinate plane
Uploaded: 2025-11-26T05:44:23.000Z
Duration: 1 min 20 s
Channel: JoVE

Dwaj przyjaciele wchodzą do parku różnymi wejściami i zatrzymują się w różnych zakątkach pola.Ze swoich pozycji chcą określić dokładną odległość między nimi.Wzór na odległość jest stosowany do obliczania odległości między dwiema pozycjami, oznaczonymi jako punkty A i B, przy użyciu ich współrzędnych.Znajduje najkrótszą drogę między nimi — jak cięcie po przekątnej pola, zamiast chodzić po krawędziach.Obliczenia te opierają się na twierdzeniu Pitagorasa, w którym odległości poziome i pionowe spotykają się w punkcie C, tworząc trójkąt prostokątny.Twierdzenie mówi, że kwadrat przeciwprostokątnej lub boku przekątnej jest równy sumie kwadratów pozostałych dwóch boków.Na siatce współrzędnych te boki reprezentują różnice w wartościach x i y między punktami A i B.Podnosząc do kwadratu różnice poziome i pionowe, dodając je do siebie i biorąc pierwiastek kwadratowy, określa się długość przeciwprostokątnej.Ta relacja daje odległość w linii prostej między dwoma przyjaciółmi.Takie obliczenia są również wykorzystywane przy wyznaczaniu tras i planowaniu zasięgu sieci bezprzewodowej.

wzór na odległość i środek odcinka

W geometrii pomiar bezpo&#347;redniej odleg&#322;o&#347;ci mi&#281;dzy dwoma punktami na p&#322;aszczy&#378;nie jest niezb&#281;dny w r&#243;&#380;nych zastosowaniach praktycznych i teoretycznych. Zar&#243;wno w nawigacji, in&#380;ynierii, jak i w grafice komputerowej wyznaczanie najkr&#243;tszej drogi mi&#281;dzy dwoma lokalizacjami wymaga u&#380;ycia wzoru na odleg&#322;o&#347;&#263;. Wz&#243;r ten wywodzi si&#281; z twierdzenia Pitagorasa, kt&#243;re wi&#261;&#380;e d&#322;ugo&#347;ci bok&#243;w tr&#243;jk&#261;ta prostok&#261;tnego. Na p&#322;aszczy&#378;nie wsp&#243;&#322;rz&#281;dnych odleg&#322;o&#347;ci pozioma i pionowa mi&#281;dzy dwoma punktami stanowi&#261; przyprostok&#261;tne tr&#243;jk&#261;ta prostok&#261;tnego, a linia &#322;&#261;cz&#261;ca te punkty tworzy przeciwprostok&#261;tn&#261;. Obliczaj&#261;c pierwiastek kwadratowy z sumy kwadrat&#243;w tych r&#243;&#380;nic wsp&#243;&#322;rz&#281;dnych poziomych i pionowych, mo&#380;na dok&#322;adnie wyznaczy&#263; odleg&#322;o&#347;&#263; w linii prostej.Planista miejski nanosi na map&#281; dwie kluczowe lokalizacje w miejscowo&#347;ci: bibliotek&#281; w punkcie L (300, 500) i kompleks sportowy w punkcie S (1000, 1200). Aby okre&#347;li&#263; odleg&#322;o&#347;&#263; mi&#281;dzy tymi lokalizacjami, stosuje si&#281; wz&#243;r na odleg&#322;o&#347;&#263;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:54.27%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40039/40039_Equation_1_1__editor_40039.svg' alt='40039_Equation_1_1_.svg' /></figure>Podstawiaj&#261;c podane wsp&#243;&#322;rz&#281;dne:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:52%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40039/40039_Equation_2_1__editor_40039.svg' alt='40039_Equation_2_1_.svg' /></figure>Zatem biblioteka i kompleks sportowy znajduj&#261; si&#281; w odleg&#322;o&#347;ci oko&#322;o 990 metr&#243;w.Ta metoda ilustruje praktyczne zastosowanie zasad geometrii w planowaniu urbanistycznym, umo&#380;liwiaj&#261;c dok&#322;adny pomiar odleg&#322;o&#347;ci, niezb&#281;dny do projektowania efektywnych uk&#322;ad&#243;w i infrastruktury.

Wzór na odległość i środek odcinka

W geometrii pomiar bezpo&#347;redniej odleg&#322;o&#347;ci mi&#281;dzy dwoma punktami na p&#322;aszczy&#378;nie jest niezb&#281;dny w r&#243;&#380;nych ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Coordinates and Graphs

Współrzędne i wykresy

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

The Distance Formula

finding distance on a coordinate plane

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings.

The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them.

To find the distance, square the horizontal change and the vertical change, add those values, and take the square root. This gives the straight-line distance between the two points. The method is used in navigation, engineering, and computer graphics, where accurate distance matters.

A city planner example shows how the formula works in real life. A library is located at L (300, 500), and a sports complex is at S (1000, 1200). Using the distance formula, the two locations are about 990 meters apart. This kind of calculation helps planners design efficient layouts and infrastructure.

Finding Distance on a Coordinate Plane

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings. The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them. <br...

the midpoint formula

Finding the Center of a Line Segment

Finding the center of a line segment is a key idea in coordinate geometry. The midpoint is the point that lies exactly halfway between two points on a coordinate plane. It is used in mathematics, physics, engineering, and planning work where a central location matters. To work with a midpoint, start with two points, A(x1, y1) and B(x2, y2). A straight line segment can be drawn between them on the coordinate plane. The midpoint, labeled M, divides that segment into two equal parts. The...

graphs of equations in two variables

Plotting Two-Variable Equations with Tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle Equations in Standard Form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

An ellipse centered at the origin shows clear symmetry in its graph. Its equation describes a smooth, closed curve whose distance from the center keeps a constant ratio between the horizontal and vertical axes. Because the ellipse stretches farther along the x-axis than the y-axis, it has a horizontal shape. This ellipse has symmetry across the x-axis. That means the top half and bottom half are mirror images across the horizontal axis. Algebraically, the equation stays the same when y is...

solving equations graphically

Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

Graphing Inequality Solution Ranges

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Learn how the distance formula uses the Pythagorean Theorem to measure straight-line distance between two points on a coordinate plane, with a city planning example.

Finding Distance on a Coordinate Plane | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Finding Distance on a Coordinate Plane

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