Lesson 3 – 6 Perpendicular and Distance Geometry Lesson 3 – 6 Perpendicular and Distance Objective: Find the distance between a point and a line. Find the distance between parallel lines.

Distance from a point to a line? Don’t copy Distance from a point to a line? 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. Perpendicular postulate If given a line and a point not on the line, then there exits exactly one line through the point that is perpendicular to the given line.

Line l contains points (-5, 3) and (4, -6) Line l contains points (-5, 3) and (4, -6). Find the distance between l and point (2, 4). Step 1: write the equation of given line Copy m = -1 y = -x - 2 Need the shortest distance from P to the line so we need the Perpendicular line from P. Step 2: Write an equation perpendicular and through the given point. m = 1 y = x + 2 Cont…

Copy Step 3: Find the point of intersection by solving the system of equations. y = -x – 2 y = x + 2 2y = 0 y = 0 0 = x + 2 -2 = x If the point of intersection can be found using the graph that is fine, but if not system of equations must be used. (-2, 0) Q Cont…

I want both for your answer! Copy Step 4: Find the distance from the given point to the point of intersection. Distance between (2, 4) and (-2, 0) Simplify! I want both for your answer!

Try Line l contains points (1, 2) and (5, 4). Find the distance between l and point P (1, 7). Step 1: write the equation of the line m = ½ 2 = (1/2)(1) + b b = 3/2 y = ½ x + 3/2 Step 2: Write an equation perpendicular and through the given point. 7 = -2(1) + b b = 9 y = -2x + 9 Cont…

Step 3: Find the point of intersection by solving the system of equations. y = ½ x + 3/2 y = -2x + 9 y = -2(3) + 9 = -6 + 9 = 3 (3, 3) x = 3 Cont…

Step 4: Find the distance from the given point to the point of intersection. (3, 3) (1, 7)

Step 1: pick a point (any point) on either line Copy Find the distance between the parallel lines l and m with equations y = 2x + 1 and y = 2x – 3, respectively. Step 1: pick a point (any point) on either line HINT – pick an easy point like a y-intercept! (0, 1) (y-intercept of first equation) Keep track of this point you will need it later! Step 2: Write an equation perpendicular and through the point from step 1. M = -1/2 (perpendicular) thru (0, 1) 1 = (-1/2)(0) + b Respectively means the equations are in the same order as the named lines.

Step 3: Use system of equations to find point of intersection Copy Step 3: Use system of equations to find point of intersection 2 equations you use Perpendicular equation The equation that you didn’t use y-intercept y = 2x – 3

Step 4:Find the distance between the two points Copy Step 4:Find the distance between the two points Point from step 1 and the point of intersection. (0, 1)

Homework Pg. 218 4 – 32 EOE