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Understanding and Constructing Perpendicular Bisectors in Geometry

In this lesson, we will explore the concept of perpendicular bisectors, learning to construct them through various methods including paper folding. Students will determine properties and perform constructions related to perpendicular bisectors. Warm-up questions and quizzes will reinforce understanding of midpoints and segments. We investigate the Perpendicular Bisector Conjecture, which states that points on the bisector are equidistant from segment endpoints, and we will apply this understanding to construct geometric shapes, promoting hands-on learning and engagement.

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Understanding and Constructing Perpendicular Bisectors in Geometry

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

  2. 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 A B 1 2 C H

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

  4. 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.

  5. 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!

  6. 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

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

  8. 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.

  9. 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.

  10. 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

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