Name Date Class Period. 5.2 Exploring Properties of Perpendicular Bisectors


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1 Name Date Class Period Activity B 5.2 Exploring Properties of Perpendicular Bisectors MATERIALS QUESTION EXPLORE 1 geometry drawing software If a point is on the perpendicular bisector of a segment, is it equidistant from the endpoints of the segment? Investigate properties of a perpendicular bisector STEP 1 Draw a line segment and perpendicular bisector Draw and label AB horizontally on the screen. Draw the perpendicular bisector of AB. Label the point of intersection of the bisector and AB as C. Locate and label a point D on the perpendicular bisector. STEP 2 Draw line segments Draw DB and DA. Measure DB and DA. Record the lengths on the screen. Drag point D along the perpendicular bisector. Note what is happening to the lengths of DB and DA. DRAW CONCLUSIONS Use your observations to complete these exercises Complete the definition of a perpendicular bisector: If DC is the perpendicular bisector of BA, then DCA and DCB are, and BC and CA are. Suppose D is a point on the perpendicular bisector of AB. What can you conclude about AD and BD? 3. If a point is on the perpendicular bisector of a segment, the distances from the point to the endpoints of the segment?. EXPLORE 2 Construct a perpendicular bisector STEP 1 Draw a line segment and point Start a new construction. Draw AB horizontally on the screen. Draw and label a point E above AB. Steps 1 and 2 1 of 5
2 STEP 2 CONSTRUCT equal LINE SEGMENTS Measure AE and BE. Record the lengths on the screen. Drag point E until BE and AE are equal. STEP 3 Construct equal line segments Step 3 Draw and label a point F below AB. Measure the length of FA and FB. Record their lengths on the screen. Drag point F until FA and FB are equal. STEP 4 Complete the construction Draw segment FE. Label the point of intersection of AB and FE point G. STEP 5 Measure line segments and angles Steps 4 and 5 Complete the table below. Measurement EGA EGB GA GB DRAW CONCLUSIONS Use your observations to answer these exercises 4. Name the perpendicular bisector of AB Explain your reasoning. 5. If EA = EB, then E is on the of AB. 6. If FA = FB, then F is on the of AB. 7. If a point is equidistant from the endpoints of a segment, then the point is on the of the segment. 2 of 5
3 Answer Key B EXPLORE 1 Investigate properties of a perpendicular bisector STEP 2 Draw line segments The lengths of DB and DA remain congruent when point D is dragged. DRAW CONCLUSIONS 3. right angles; congruent They are congruent. The distances are equal EXPLORE 2 Construct a perpendicular bisector STEP 5 Measure line segments and angles Answers will vary. DRAW CONCLUSIONS 4. EF is the perpendicular bisector of AB because it is perpendicular to AB at the midpoint of AB. 5. perpendicular bisector 6. perpendicular bisector 7. perpendicular bisector 3 of 5
4 Teacher Notes ACTIVITY PREPARATION AND MATERIALS Geometry drawing software on a graphing calculator or computer Overhead projector and transparencies (optional) It is not necessary that each student have a calculator or computer. This activity can be done in pairs. ACTIVITY MANAGEMENT You may limit the time the activity takes by only doing one of the two Explores. Students will need to know how to use the geometry drawing software to draw and measure line segments, draw a perpendicular bisector, and select and move points. Common Error In Explore 1, it is important the point D is constructed on the perpendicular bisector. Students should not create a point and move it onto the perpendicular bisector. 4 of 5
5 Activity and Closure Questions Place the diagram below on the board or overhead and ask the following question. In the diagram, XZ is the perpendicular bisector of YT. List all valid statements you can make. Answer: XZ YT, XZY = XZT = 90, YZ = ZT, Z is the midpoint of YT, Y X = XT, Δ XYT is isosceles, Δ YXZ Δ TXZ, YXZ = TXZ, XYZ = XZT, Δ XYZ is a right triangle, Δ XZT is a right triangle Place the diagram below on the board or overhead and ask the following questions. a. If RH = RN then R is on the of H N. Answer: perpendicular bisector b. If HJ = J N then J is on the of H N. Answer: perpendicular bisector 3. Explain how Exercises 3 and 7 are related. Answer: They are converses of each other LESSON TRANSITION In Lesson 5.2, the perpendicular bisector theorem and its converse are introduced. This activity is designed to introduce the theorem and its converse. Students will use the theorem and its converse to solve algebraic equations. In addition, students will investigate the point of concurrency of the perpendicular bisectors of a triangle. 5 of 5
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