GEOMETER'S SKETCHPAD
ASSIGNMENT #4 (Youngberg)
Note the following instructions as you complete this and all other GSP assignments:
1.
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Although you will be emailing your assignment to me, save a copy of all of your sketches to your disk.
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Answer all of the questions posed to you in a text box at the bottom of the sketch you are currently working on.
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Make sure that all of your sketches are constructions, not drawings. That is, create all sketches so that the required
geometric relationships are maintained when any object is “dragged.”
In section 12.1 of the text, we learned that two figures are congruent if one is the image of the other under an isometry (recall
that an isometry is a transformation that preserves size and shape). In particular, if two figures are congruent, then one is the
image of the other due to a single reflection, a single rotation, a single translation, or a single glide reflection.
In the figure to the right, the first image is the result of reflecting the
original figure across the first mirror. The second image is the result of
reflecting the first image across the second mirror. The second mirror is
constructed so that it is perpendicular to the first mirror.
In a new sketch, construct an original figure that looks like the one
shown. Then construct an arbitrary line to be your first mirror and a line
perpendicular to that line to be your second mirror. Perform the
reflections described above.
first image
original figure
first mirror
Now, Hide everything except for the original figure and the second
image. Notice that the second image is congruent to the original figure.
Thus, there must be a single isometry that will send the original figure
onto the second image.
Determine whether the original figure can be sent onto the second image
by a reflection (if so, find and label the mirror line) or a rotation (if so,
find and label the center and the rotation angle) or a translation (if so,
find and label the translation vector) or a glide reflection—a combination
of a translation and a reflection— (if so, find and label the translation
vector and the mirror line.)
second mirror
second image
Test your results by constructing the interior of the original figure and
sending it (just the interior) onto the second image with your chosen
isometry.
QUESTION 1:
If a single reflection was used, how does the line of reflection relate to the initial two mirror lines? If a
single rotation was used, how do the center and angle of rotation relate to the initial two mirror lines? If a
translation was used, how does the directed segment relate to the initial two mirror lines? If a glide
reflection was used, how do the directed segment and the glide axis relate to the initial two mirror lines?
Save this sketch as GSP 4A (find single isometry).
2.
In a new sketch, do the attached handout titled “Properties of Reflection.” Be sure to answer all questions posed to you in the
handout in a textbox at the bottom of your sketch.
Save your sketch as GSP 4B (properties of reflection).
3.
In a new sketch, construct a triangle. Label the vertices A, B, and C. Construct a point near (but not on) the triangle and
label that point “center” with the label tool. Mark that point as a center. Somewhere out of the way, construct two segments
(their lengths may be different.) Measure the ratio of the lengths of the segments by selecting them together and choosing
Ratio in the Measure menu. Mark this ratio as the scale factor for a dilation by selecting the measurement and choosing
Mark Scale Factor in the Transform menu.
Select all sides and vertices of ∆ABC and dilate them by the marked ratio. Use the label tool to display the labels (A’, B’,
and C’) of the dilated image. Experiment with your sketch by changing the size and shape of ∆ABC, relocating the center,
and changing the scale factor by changing the lengths of the segments that you marked as the ratio.
Determine the relationship between the center point, the point A, and the point A’ (not in terms of distance, but in terms of
where they are in relationship to each other.) Perform sufficient constructions to confirm this relationship.
QUESTION 1:
Describe the relationship just determined.
Determine the relationship between the segments
confirm this relationship.
QUESTION 2:
AB and A ' B ' . Perform sufficient constructions and/or measurements to
Describe the relationship just determined.
Save your sketch as GSP 4C (dilation exploration).
4.
Open the sketch GSP 4C (dilation exploration)from the previous problem (if it isn’t already open). Immediately save this
sketch again as GSP 4D (dilation areas).
Construct the interior of ∆ABC and of its image, ∆A’B’C’. Measure the area of ∆ABC and of its image. Calculate the ratio
area (∆A ' B ' C ')
using the calculator.
area ( ABC )
Change the original triangle, the center of dilation, and the scale factor by dragging the appropriate objects with the mouse.
As you make these changes, observe how the ratio of the areas changes in relation to the ratio that you marked as the scale
factor.
Determine specifically what this relationship is. (Hint: To help you discover this relationship, you may wish to make a table
on scratch paper and record different values for the scale factor in one column and the corresponding ratio of the areas in the
other. Try setting the scale factor to whole numbers, i.e. to 2, then 3, then 4, then 5, etc. and looking for a pattern. Ignore the
rounding error in GSP’s measurements.)
QUESTION 1:
Describe the relationship just determined.
Recall that you have already saved this sketch as GSP 4D (dilation areas). Once you have completed this problem to your
satisfaction, don’t forget to save the changes.
5.
In a new sketch, construct the following image using reflections,
rotations, and size-transformations. (Hint: To construct an arc on a
circle, select the endpoints of the desired arc in counter-clockwise order
and select the circle, then choose Arc on Circle from the Construct
menu. The circle will obscure the arc until you hide the circle.) Make
sure that you label a point that controls the size of the web.
Save your sketch as GSP 4E (web).
controls size of web