Finding the coordinates of a segment. How to calculate the length of a segment from given coordinates

By segment call a part of a straight line consisting of all points of this line that are located between these two points - they are called the ends of the segment.

Let's look at the first example. Let a certain segment be defined by two points in the coordinate plane. In this case, we can find its length by applying the Pythagorean theorem.

So, in the coordinate system we draw a segment with the given coordinates of its ends (x1; y1) And (x2; y2) . On axis X And Y Draw perpendiculars from the ends of the segment. Let us mark in red the segments that are projections from the original segment on the coordinate axis. After this, we transfer the projection segments parallel to the ends of the segments. We get a triangle (rectangular). The hypotenuse of this triangle will be the segment AB itself, and its legs are the transferred projections.

Let's calculate the length of these projections. So, onto the axis Y projection length is y2-y1 . and on the axis X projection length is x2-x1 . Let's apply the Pythagorean theorem: |AB|² = (y2 - y1)² + (x2 - x1)² . In this case |AB| is the length of the segment.

If you use this diagram to calculate the length of a segment, then you don’t even have to construct the segment. Now let’s calculate the length of the segment with coordinates (1;3) And (2;5) . Applying the Pythagorean theorem, we get: |AB|² = (2 - 1)² + (5 - 3)² = 1 + 4 = 5 . This means that the length of our segment is equal to 5:1/2 .

Consider the following method for finding the length of a segment. To do this, we need to know the coordinates of two points in some system. Let's consider this option, using a two-dimensional Cartesian coordinate system.

So, in a two-dimensional coordinate system the coordinates are given extreme points segment. If we draw straight lines through these points, they must be perpendicular to the coordinate axis, then we get right triangle. The original segment will be the hypotenuse of the resulting triangle. The legs of a triangle form segments, their length is equal to the projection of the hypotenuse on the coordinate axes. Based on the Pythagorean theorem, we conclude: in order to find the length of a given segment, you need to find the lengths of the projections onto two coordinate axes.

Let's find the projection lengths (X and Y) the original segment onto the coordinate axes. We calculate them by finding the difference in the coordinates of points along a separate axis: X = X2-X1, Y = Y2-Y1 .

Calculate the length of the segment A . To do this, find the square root:

If our segment is located between points whose coordinates 2;4 And 4;1 . then its length is correspondingly equal to √((4-2)²+(1-4)²) = √13 ≈ 3.61 .

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how to calculate the length of a segment by given coordinates: Length of a segment A segment is a part of a straight line consisting of all points of this line that are located between these two points - they are called the ends of the segment. Let's consider

There are three main coordinate systems used in geometry, theoretical mechanics, and other branches of physics: Cartesian, polar and spherical. In these coordinate systems, each point has three coordinates. Knowing the coordinates of two points, you can determine the distance between these two points.

You will need

Cartesian, polar and spherical coordinates of the ends of a segment

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Instructions

Consider first a rectangular Cartesian coordinate system. The position of a point in space in this coordinate system is determined x,y coordinates and z. A radius vector is drawn from the origin to the point. The projections of this radius vector onto the coordinate axes will be the coordinates of this point.
Let you now have two points with coordinates x1,y1,z1 and x2,y2 and z2 respectively. Denote by r1 and r2, respectively, the radius vectors of the first and second points. Obviously, the distance between these two points will be equal to the magnitude of the vector r = r1-r2, where (r1-r2) is the vector difference.
The coordinates of the vector r will obviously be: x1-x2, y1-y2, z1-z2. Then the magnitude of the vector r or the distance between two points will be equal to: r = sqrt(((x1-x2)^2)+((y1-y2)^2)+((z1-z2)^2)).

The polar coordinates of a point can be converted to Cartesian coordinates as follows: x = r*cos?, y = r*sin?, z = z. Then the distance between two points with coordinates r1, ?1 ,z1 and r2, ?2, z2 will be equal to R = sqrt(((r1*cos?1-r2*cos?2)^2)+((r1*sin? 1-r2*sin?2)^2)+((z1-z2)^2)) = sqrt((r1^2)+(r2^2)-2r1*r2(cos?1*cos?2+sin ?1*sin?2)+((z1-z2)^2))

Now consider a spherical coordinate system. In it, the position of a point is specified by three coordinates r, ? And?. r is the distance from the origin to the point, ? And? - azimuth and zenith angle, respectively. Corner? similar to an angle with the same designation in the polar coordinate system, eh? - the angle between the radius vector r and the Z axis, with 0

Let's convert spherical coordinates to Cartesian: x = r*sin?*cos?, y = r*sin?*sin?*sin?, z = r*cos?. The distance between points with coordinates r1, ?1, ?1 and r2, ?2 and?2 will be equal to R = sqrt(((r1*sin?1*cos?1-r2*sin?2*cos?2)^2 )+((r1*sin?1*sin?1-r2*sin?2*sin?2)^2)+((r1*cos?1-r2*cos?2)^2)) = ((( r1*sin?1)^2)+((r2*sin?2)^2)-2r1*r2*sin?1*sin?2*(cos?1*cos?2+sin?1*sin?2 )+((r1*cos?1-r2*cos?2)^2))

How simple

You will need

Cartesian, polar and spherical coordinates of the ends of a segment

Instructions

Consider first a rectangular Cartesian coordinate system. The position of a point in space in this coordinate system is determined coordinates x,y and z. A radius vector is drawn from the origin to the point. The projections of this radius vector onto the coordinate axes will be coordinates this point.
Suppose you now have two points with coordinates x1,y1,z1 and x2,y2 and z2 respectively. Denote by r1 and r2, respectively, the radius vectors of the first and second points. Obviously, the distance between these two points will be equal to the magnitude of the vector r = r1-r2, where (r1-r2) is the vector difference.
The coordinates of the vector r will obviously be: x1-x2, y1-y2, z1-z2. Then the magnitude of the vector r or the distance between two points will be equal to: r = sqrt(((x1-x2)^2)+((y1-y2)^2)+((z1-z2)^2)).

Consider now a polar coordinate system, in which the coordinate of a point will be given by the radial coordinate r (radius vector in the XY plane), the angular coordinate? (the angle between the vector r and the X axis) and the z coordinate, similar to the z coordinate in the Cartesian system. The polar coordinates of a point can be converted to Cartesian coordinates as follows: x = r*cos?, y = r*sin?, z = z. Then the distance between two points with coordinates r1, ?1 ,z1 and r2, ?2, z2 will be equal to R = sqrt(((r1*cos?1-r2*cos?2)^2)+((r1*sin?1-r2*sin?2 )^2)+((z1-z2)^2)) = sqrt((r1^2)+(r2^2)-2r1*r2(cos?1*cos?2+sin?1*sin?2) +((z1-z2)^2))

Now consider a spherical coordinate system. In it, the position of the point is specified by three coordinates r, ? And?. r is the distance from the origin to the point, ? And? - azimuth and zenith angle, respectively. Corner? similar to an angle with the same designation in the polar coordinate system, eh? - the angle between the radius vector r and the Z axis, and the coordinates r1, ?1, ?1 and r2, ?2 and ?2 will be equal to R = sqrt(((r1*sin?1*cos?1-r2*sin? 2*cos?2)^2)+((r1*sin?1*sin?1-r2*sin?2*sin?2)^2)+((r1*cos?1-r2*cos?2) ^2)) = (((r1*sin?1)^2)+((r2*sin?2)^2)-2r1*r2*sin?1*sin?2*(cos?1*cos?2 +sin?1*sin?2)+((r1*cos?1-r2*cos?2)^2))

Let a segment be defined by two points in the coordinate plane, then its length can be found using the Pythagorean theorem.

Instructions

Let the coordinates of the ends of the segment (x1- y1) and (x2- y2) be given. Draw a line segment in the coordinate system.

Draw perpendiculars from the ends of the segment on the X and Y axes. The segments marked in red in the figure are projections of the original segment on the coordinate axes.

If you do parallel transfer, projection segments to the ends of the segments, you get a right triangle. The legs of this triangle will be the transferred projections, and the hypotenuse will be the segment AB itself.

Projection lengths are easy to calculate. The length of the projection on the Y axis will be equal to y2-y1, and the length of the projection on the X axis will be x2-x1. Then, by the Pythagorean theorem, |AB|²- = (y2 - y1)²- + (x2 - x1)²-, where |AB| - length of the segment.

Having presented this scheme for finding the length of a segment in general case, it is easy to calculate the length of a segment without constructing a segment. Let's calculate the length of the segment whose end coordinates are (1-3) and (2-5). Then |AB|²- = (2 - 1)²- + (5 - 3)²- = 1 + 4 = 5, so the length of the required segment is 5^1/2.

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