A coordinate plane

A coordinate plane consists of (x, y) point pairs. In the coordinate system, the x-axis is horizontal and the y-axis is vertical. The point of intersection of the axes, i.e. the point (0,0), is called the origin. The coordinate system is already familiar from the previous course.

The distance between two points, i.e. the length of the line segment between the points.

Example 1

Find the distance between points (2,3) and (5,7).

In the coordinate system we can form a right triangle with the points. The length of the hypotenuse of a triangle is the distance between the points. In the figure, the hypotenuse is denoted with the letter h.

So we can solve the distance between the points using Pythagorean theorem. Since it is a length, only a positive solution is valid. The distance between the points is 5.

The formula for the distance between points found in the tables book comes from Pythagorean theorem.

Distance between points (x₁, y₁) and (x₂, y₂) (line segment length)

Example 2

Find the midpoint of the line segment when the endpoints of the line segment are (0,2) and (6,6)

The midpoint of the line segment is found in the mid-horizontal as well as the mid-vertical distance between the endpoints. The difference between the x-coordinates is 6, so the horizontal distance of the centre from the starting point is 3. The difference between the y-coordinates is 4, so the vertical distance from the starting point is 2. Thus, the centre is (3,4). The midpoint of the line segment is the arithmetic mean of the endpoints.

In general, the midpoint of a line segment is obtained by knowing the endpoints (x₁, y₁) and (x₂, y₂)

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