Chapter 3.6 Perpendiculars and Distance

Chapter 3.6 Perpendiculars and Distance Check.4.8 , Apply properties and theorems about angles associated with parallel and perpendicular lines to solve problems. CLE 3108.3.1 Use analytic geometry tools to explore geometric problems involving parallel and perpendicular lines, circles, and special points of polygons.

___ ___ A.AB || CD B.FG || HI C.CD || FG D. none __ ___ ___ ___ Given 9  13, which segments are parallel?

Two lines in the same plane do not intersect. Which term best describes the relationship between the lines? A. parallel B. perpendicular C. skew D. transversal

Question • What is the quickest way to get from this classroom to the cafeteria? • If you are on the playing field or court, what is the quickest route to leave the field?

Construction of Parallel Line through a Point • Draw a Line, two points M, N • Draw point P not on the line. • Draw PM • Copy PMN so P is the vertex of the new angle • Know that PMN  RPQ then MN  PQ R P Q N M

Construction of a Perpendicular Segment l • Use Graph Paper. Line l goes through points (-6, -9) and (0,-1) • Construct a line perpendicular to line l through P (-7, -2) • Put the compass at P and draw an arc intersecting l in two places, label these A and B • Put the compass at A and drawn an arc below line l (must be ½ length of AB) • Using same setting put the compass at point B and draw and arc intersecting the previous one. • Label the intersection Q P B A Q • Draw PQ. Check the slope to verify perpendicularity

Construction of a Perpendicular Segment l • Use the distance formula to calculate the distance between P and line l . • (-7, -2) P • (-3, -5) intersection P B A Q

Construction of a Perpendicular Segment V • Use Graph Paper. • Construct a line perpendicular to line s (y = -x) through V (1,5) • Find the Distance from V to s • Distance is 18

Distance between Point and a Line • The shortest distance between a point and a line is the length of the segment perpendicular to the line from that point C A B

Distance from Point to a line C B A

Distance between Parallel Lines • Two parallel lines are equidistant from each other. The distance between two parallel lines is the distance between one of the lines and any point on the other line l m n • Theorem – In a plane, if two lines are equidistance from a third, then the two lines are parallel to each other

Distance between two parallel lines 3 8

Distance between two lines • Find the distance between the parallel lines l and m whose equations are y = -1/3x -3 and y = -1/3x+ 1/3 respectively. • Draw perpendicular line • Find the point where the two lines cross. • Calculate Distance between points • Solve system of equations • Write equation for line perpendicular passing through line • m = 3, point (0, -3) • y – y1 = m (x – x1) • y –(-3)=3(x – 0) • y +3 = 3x • y = 3x - 3

Find the intersection of the two lines or the solution to the system of equations y = 3x – 3 and y = -1/3x+ 1/3 Substitute for y 3x – 3 = -1/3x+ 1/3 +1/3 x + 3 = +1/3x +3 3 1/3x = 3 1/3 x = 1 Substitute x into either equation y = 3(1) – 3 y = 0 Calculate distance between the two points (0, -3) and (1, 0)

Find the distance between the parallel lines • y = 2x +3 and y = 2x – 3 • m = -1/2 point (0, -3) • y – (-3) = -1/2(x – 0) • y + 3 = -1/2x • y = -1/2x -3 and y = 2x + 3 • -1/2x -3 = 2x + 3 • +1/2x - 3 + 1/2x - 3 • -6 = 2.5x • x= -2.4 , y = 2(-2.4)+3=-1.8 • (0, -3) and (-2.4, -1.8)

Practice Assignment • Block Page 218, 10 - 28 Even

Chapter 3.6 Perpendiculars and Distance