Constructing Perpendicular Bisectors During this lesson, we will: Construct the perpendicular bisector of a segment Determine properties of perpendicular bisectors

Daily Warm-Up Quiz A point which divides a segment into two congruent segments is a(n) _____. 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. Mark the figure based upon the given information: a. Angle 2 is a right angle. b. H is the midpoint of BC H C B A 1 2

I wonder how many segment bisectors I can draw through the midpoint? Before we start: a line, segment, or ray which intersects a segment at its midpoint Segment Bisector: ________________ ______________________________ 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 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. 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. Z Y X

Investigative Results: Perpendicular Bisector Conjecture If a point lies on the perpendicular bisector of a segment, then it is _______ from each of the endpoints. Converse: If a point is equidistant from the endpoints of a segment, then it is on the __________________. perpendicular bisector equidistant 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.

Final Checks for Understanding Construct the “average” of HI and UP below. _______________ _______ H I U P 2. Name two fringe benefits of constructing perpendicular bisectors of a segment. 3.* By construction, divide a segment into fourths (four congruent segments). 4. BONUS: Construct a segment which is ¾ the length of your original segment in #3 above.

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