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CC Geometry 10R Aim 9: How do we construct a perpendicular bisector? 18° Do Now: 62 1. Complete: An angle bisector is a ray (line/segment) angle that divides an ________ into = or ≅ parts. two ___ 2. Construct and label BD, the bi sector of ≮ABC. D A B C Relevant Vocabulary Two lines (segments, rays) are PERPENDICULAR ( Right _________________ angle. Ex. a) Draw and label: AB ) if they intersect to form a CD , AB and CD intersect at E. C b) Name all right angles in the diagram you drew. <AEC <BEC <AED E <DEB A B D congruent The MIDPOINT of a segment is a point that divides a segment into 2 = or _____ parts. Ex. Draw and label: L is the midpoint of XY. x y L midpoint A SEGMENT BISECTOR passes through the ____________ of a segment. A Ex. Draw and label: AB is a segment bisector of XY, but AB is not to XY. X Y B The PERPENDICULAR BISECTORof a line segment is perpendicular to a segment at its midpoint. Ex. Draw and label: AB is the perpendicular bisector XY; M is the midpoint of XY. A X M B Y CONSTRUCTION #4: CONSTRUCT THE PERPENDICULAR BISECTOR OF A LINE SEGMENT Experiment with your construction tools to establish a construction that results in the perpendicular bisector. [Hint: Use what you know about constructing an http://www.mathsisfun.com/geometry/constructlinebisect.html equilateral triangle.] Steps 1. Draw circle(arc) A, center A, radius more than ½AB. 2. Draw circle(arc) B, center B, using same radius as in step (1). A 3. Label the points of intersection created by steps (1) and (2) as C and D. B 4. Draw CD. Practice: Construct the perpendicular bisector of CD. • Label the midpoint M and labelthe perpendicular bisector as EF. <EMC • Name one right angle:__________. C M D A point is EQUIDISTANT from two given points if it is the same distance from both given points. C Ex a) Draw point C equidistant from points A and B. b) Draw point D equidistant from points A and B. c) Draw point E equidistant from points A and B. d) How many points could you draw equidistant from A and B? A D B E Activity: CDE is the perpendicular bisector of AB. (C, D, and E are collinear.) Using your compass, what conclusion can you make about the following pairs of segments? Each point is Equidistant 1) AC and BC from points A and B 2) AD and BD 3) AE and BE Based on your findings, fill in the observation below. Any point on the perpendicular bisector of a line Equidistant segment is _____________________ from the endpoints of the line segment. CONSTRUCTION #5: CONSTRUCT A PERPENDICULAR TO A LINE FROM A POINT NOT ON THE LINE line to line l from a point X not on line l . We will now construct a perpendicular Complete the construction and the steps of the construction outlined below. X l C B D Step 1: Step 2: Step 3: Step 4: Step 5: Draw circle(arc) X so that the circle intersects line l in two points. Label the two points of intersection as B and C. Draw circle(arc) B and circle(arc) C, same radius as Step 1. Label the intersection of circle B and circle C as D . XD Draw the perpendicular bisector: line _______. Exercises 1. Divide segment AB into 4 segments of equal length. B A 2. Construct parallel lines l1 and Step 1: Construct line Step 2: Construct line l2 as follows: l3 which will be perpendicular to line l1 from point A l2 which will be perpendicular to l3 through point A. (Hint: This is the same as bisecting a straight angle.) A l1 3. Here is another method for constructing a line parallel to a given line through a point not on the line, not using perpendicular lines. Using the construction for copying an angle, construct a line parallel to line L through point P. P L 4a) Construct the perpendicular bisector of BC. b) Construct the angle bisector of ≮B. C A B Let's Sum it Up!! A perpendicular bisector of a segment passes midpoint through the ______________of the segment Right angles with the segment. and forms ________ (Mark the diagram to show this.) A A point A is said to be equidistant from two different points B and C if AB = AC. (Mark the diagram to show this.) B C Construction of a perpendicular to a line from a point not on the line - arcs (not full circles) A step 3 l step 1 step 2 step 2