8.1 The Rectangular Coordinate System and Circles

Part 1: Distance and Midpoint Formulas

Rectangular Coordinates

• A pair of numbers in the form (x , y) is an example of an ordered pair.

• The numbers in an ordered pair are the components of the ordered pair.

• An ordered pair is plotted (or graphed) on a rectangular (or Cartesian) coordinate system.

• The axes from four quadrants. (A point on an axis is not considered to be in any quadrant.)

x-axis

y-axis

I

IIIII

IV

origin

Distance Formula

• The distance between the points (x1, y1) and (x2, y2) is

2 2

2 1 2 1d x x y y

Finding the Distance Between Two Points

• Find the distance between (-3, 5) and (6, 4).

Find the distance between (3, 4) and (-2, 1).

Midpoint Formula

• The midpoint of a line segment is the point on the segment that is equidistant from both endpoints.

• Given the coordinates of the two endpoints of a line segment, we can find the midpoint by “averaging” the coordinates…or by using

1 2 1 2,2 2

x x y yM

Finding the Midpoint of a Segment

• Find the coordinates of the midpoint of the line segment with endpoints (8, -4) and (-9, 6).

Find the coordinates of the midpoint of the line segment with endpoints (3, 4) and (-2, 1).

Graphing in the Third Dimension(NOT in the book!)

• A 3-dimensional graph has a z-axis along with the x- and y-axis, so points have 3 coordinates: (x, y, z).

3-D Distance Formula

3-D Midpoint Formula

2 2 2

2 1 2 1 2 1d x x y y z z

1 2 1 2 1 2, ,2 2 2

x x y y z zM