AMSCO'S Geometry. New York - Rye High School

Equidistant Lines in Coordinate Geometry 621Equidistant from Two Parallel LinesThe locus of points equidistant from two parallel lines is a line parallel to thetwo lines and midway between them.For example, the locus of points equidistant from the vertical lines x 2and x 6 is a vertical line midway between them. Since the given lines intersectthe x-axis at (2, 0) and (6, 0), the line midway between them intersects thex-axis at (2, 0) and has the equation x 2.EXAMPLE 2Write an equation of the locus of points equidistant from the parallel linesy 3x 2 and y 3x 6.SolutionThe locus is a line parallel to the given lines andmidway between them.The slope of the locus is 3, the slope of the givenlines.The y-intercept of the locus, b, is the average ofthe y-intercepts of the given lines.Oy1y = 3x 2y = 3x 6y = 3x 21xb 2 1 (26)2 5 242 522The equation of the locus is y 3x 2. AnswerNote that in Example 2, we have used the midpoint of the y-intercepts of thegiven lines as the y-intercept of the locus. In Exercise 21, you will prove that themidpoint of the segment at which the two given parallel lines intercept they-axis is the point at which the line equidistant from the given lines intersectsthe y-axis.Equidistant from Two Intersecting LinesThe locus of points equidistant from two intersecting lines is a pair of lines thatare perpendicular to each other and bisect the angles at which the given linesintersect. We will consider two special cases.1. The locus of points equidistant from the axesThe x-axis and the y-axis intersect at the originto form right angles. Therefore, the lines thatbisect the angles between the axes will also gothrough the origin and will form angles measuring45° with the axes. One bisector will have a positiveslope and one will have a negative slope.Oy45°B(a, a)A(a, 0)xB(a, a)