Judith Coutts
Westwood Regional Jr./Sr. High School

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OBSERVATION WORKSHEET for DEFINITION OF REFLECTION

NAME____________________________CLASS________________________________ Write the definition of reflection (Page 157) _____________________ __________________________________________________________

Using a protractor and a straight edge, draw the following using the Givens and the steps outlined for an automatic draw program for the computer.

Given:

1. Point [A]
2. Point [B]
3. Point [C]

Steps:

1. Let [j] = Line thru Point [B] and Point [A].
2. Construct Point [C-Prime] by reflecting Point [C] through Segment [j].
3. Let [D] = random point on line [j].
4. Construct Point [D-Prime] by reflecting point [D] through line [j].

Construction using GSP

On Display choose Preferences.... Choose AUTOSHOW for Points and Straight Objects . Choose Length - Cm and Precision Hundredths; Angles - Degrees and Precision Hundredths; .

On File choose New Sketch. Again on File Choose Open Choose Def.Reflection (a script).

In order to play the script, you need the POINT Tool to DRAW three noncollinear points on the sketch. SELECT all three points, then PLAY Script. (You will have to match your points, A, B and C with the points of the script).

Compare the computer screen with your drawing. Describe the differences .___________________________________

If this construction is correct according to the definition of reflection, what can we check to verify the consequent (conditions) of the definition?

1. If a point not on the line is reflected, then the reflecting line is perpendicular to the segment connecting the preimage point with the reflection image point.

   According to the definition of perpendicular (page 132): If two lines are perpendicular, then _____________________________

   Use the computer to check this.

   • Select points C and C-Prime and Construct segment
   • Select this segment and line j and Construct point at intersection
   • select the three points of the angle and Measure angle.

   Is this consequent of the definition of reflection verified by the computer sketch? _______

2. If a point not on the line is reflected, then the reflecting line bisects the segment connecting the preimage point with the reflection image point.

   According to the definition of bisector of a segment (page 112): If a line bisects a segment, then _____________________________

   Use the computer to check this.

   • Select point C and point at intersection and Measure distance.
   • Select point C-Prime and point at intersection and Measure distance.

   Is this consequent of the definition of reflection verified by the computer sketch? __________

   DRAG point C around the sketch - Do these conditions hold true for all variations of the construction? __________________

3. If a Point is on the reflecting line then the reflection of this point is the point itself. D was constructed on line AB.

To show that point D is the same point as point D-Prime, we demonstrate that:
1. D-Prime is on line AB by using the betweenness theorem (page 41) which says ________________________________________ and then to show that
2. D Prime is the same point we will use the unique distance postulate (Page 35) which says _______________________________________________

Use the computer to check this:

• Select A and D and Measure distance .
• Select D and B and Measure distance .
• Select A and B and Measure distance . Is D between A and B?________
• Select A and D-Prime and Measure distance .
• Select D-Prime and B and Measure distance. Is D-Prime between A and B? _______

Does AD = AD Prime? ________ If so, D and D-Prime must be the same point.

We have used the computer to verify the three conclusions in the consequent of the Definition of Reflection.

NOTES TO TEACHER:

To introduce students to the script function. The simple script to reflect a point must be available to students. Instructions to animate the construction and observe the relationships under a variety of conditions are included here.

To Animate point C on a Hidden line.

With the SEGMENT Tool, DRAW a four inch segment that intersects the reflecting line near the midpoint at about a 45 degree angle. SELECT this segment and point C.

One the Edit Menu Choose Action Button-Animation... Choose Bidirectional and Medium Speed. MATCH Point C to the Segment. ANIMATE.

Now CLICK on the sketch to stop the animation. Select the segment and both of its endpoints. On Display choose Hide Objects.

DOUBLE CLICK the ANIMATE button. With the SHIFT held down during the animation, the measures will change verifying the definition under all conditions.