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Constructing Perpendicular Bisectors During this lesson, we will: Construct the perpendicular bisector of a segment Determine properties of perpendicular bisectors Daily Warm-Up Quiz 1. A point which divides a segment into two congruent segments is a(n) _____. 2. If M is the midpoint of AY, then a. AM = MY c. Both a and b. b. AM + MY = AY d. Neither a nor b. 3. Mark the figure based upon the given A information: a. Angle 2 is a right angle. b. H is the midpoint of BC B 1 H 2 C Before we start: a line, segment, or ray Segment Bisector: ________________ which intersects a segment at its midpoint ______________________________ I wonder how many segment bisectors I can draw through the midpoint? Paper-Folding a Perpendicular Bisector STEP 1 Draw a segment on patty paper. Label it OE. STEP 2 Fold your patty paper so that the endpoints O and E overlap with one another. Draw a line along the fold. STEP 3 Name the point of intersection N. Next, measure a. the four angles which are formed, and b. segments ON and NE. Definition: Perpendicular Bisector Perpendicular bisector: _____________ a line, ray, or _______________________________ segment that a. intersects a segment at _______________________________ its midpoint and b. forms right angles (90) Add each definition to your illustrated glossary! Investigation 1: Perpendicular Bisector Conjecture STEP 1 Pick three points X, Y, and Z on the perpendicular bisector. Z Y X STEP 2 From each point, draw segments to each of the endpoints. STEP 3 Use your compass to compare the following segment: a.) AX and BX, b.) AY and BY, and c.) AZ & BZ. Investigative Results: Perpendicular Bisector Conjecture If a point lies on the perpendicular equidistant bisector of a segment, then it is _______ from each of the endpoints. Shortest distance measured here! Construction: Perpendicular Bisector, Given a Line Segment Absent from class? Click HERE* for step-by-step construction tips. Please note: This construction example relies upon your first constructing a line segment. Construction: Perpendicular Bisector, Given a Line Segment Converse: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector __________________. Final Checks for Understanding 1. Construct the “average” of HI and UP below. _______________ _______ H I U P 2. Name two fringe benefits of constructing perpendicular bisectors of a segment. ENRICHMENT Now that you can construct perpendicular bisectors and the midpoint, you can construct rectangles, squares, and right triangle. Try constructing the following, based upon their definitions. Median: Segment in a triangle which connects a vertex to the midpoint of the opposite side Midsegment: Segment which connects the midpoints of two sides of a triangle