Overview of Section 3.3 The Distance formula
Theorem 3.6 The Distance formula
Given a line AB that is not parallel to either axis, the proof of the theorem follows immediately from use of the coordinate system and the Pythagorean Theorem.
The Equation of a circle
A circle is a set of points P(x,y) a fixed distance R from a given point C(x,y). Using the distance formula we have
The Equation of a Line
To develop an equation for a line from the distance formula, notice that any line can be be defined as the perpendicular bisector of some segment CD. Then use the distance formula because P(x,y) on the line is equidistant from the endpoint of the segment.
Equation of a Line Given Two Points on the Line
Slopes of Perpendicular lines
Theorem 3.7: If two lines (neither of which are vertical) are perpendicular, then the product of their slopes is - 1.
Converse: If the product of the slopes of two lines is -1, then the lines are perpendicular.
Corollary: Two lines are parallel if and only if they have the same slopes
PROBLEMS: [in the following there is no loss of generality in selecting A and B in the coordinate system with A(0,0) and B(1,0)]
a. Given two points A and B in the plane, find the locus of all points P in the plane that are twice as far from A as from B.
b. Generalize. Given two points A and B in the plane, find the locus of all points P in the plane that are m times as far from A as from B.
c. Contrast your equations for m = 2, m = 1, and m = 1/2.
PROBLEM SET 3.3