3-6 Perpendiculars and Distance Ms. Andrejko

Real World

Vocabulary Equidistant- the distance between two lines measured along a perpendicular line to the lines is always the same The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point. The distance between two parallel lines is the perpendicular distance between one of the lines and any point on the other line.

Theorems & Postulates Postulates Theorem 3.6 3.9 – Two Lines Equidistant from a Third- in a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other.

Examples Construct the segment that represents the distance indicated.

Examples Construct the segment that represents the distance indicated.

Steps (when given line and pt.) 1. Find perpendicular slope 2. Make equation using perpendicular slope and given point (P) 3. Solve system of equations to find point of intersection 4. Plug into distance formula

Examples Find the distance from P to ℓ. Line ℓ y=-x-2. Point P has coordinates (2, 4) Perpendicular slope: Equation through point: System of equations: Distance formula: m=1 y-4 = 1(x-2) y-4 = x-2 y=x+2 -x-2=x+2 -2=2x+2 -4=2x -2=x y=x+2 y=-2+2 y=0 (-2,0) √(-2-2)2+(0-4)2 √(-4)2+(-4)2 √16+16 √32 = 4√2

Practice Find the distance from P to ℓ. Line ℓ y=-x. Point P has coordinates (1, 5) Perpendicular slope: Equation through point: System of equations: Distance formula: m=1 y-5 = 1(x-1) y-5 = x-1 y= x+4 -x = x+4 -2x=4 x=-2 y=-2+4 y=2 (-2,2) √(-2-1)2+(2-5)2 √(-3)2+(-3)2 √9+9 √18 = 3√2

Example Find the distance from P to ℓ. Line ℓ y=(4/3)x-2. Point P has coordinates (-1, 5) Perpendicular slope: Equation through point: System of equations: Distance formula:

Practice Find the distance from P to ℓ. Line ℓ y=-3x+8. Point P has coordinates (-1,1) Perpendicular slope: Equation through point: System of equations: Distance formula:

Steps (when given 2 parallel equations) 1. Find y-intercept of line m 2. Write equation of perpendicular line through y-intercept (line p). 3. Use systems of equations to determine point of intersection between l and p. 4. Use distance formula

Examples Find the distance between each pair of parallel lines with the given equations. y=5x-22 y=5x+4

Practice Find the distance between each pair of parallel lines with the given equations. y=-3x+3 y=-3x-17