GEOMETER'S SKETCHPAD
ASSIGNMENT #2 (Youngberg)
Note the following instructions as you complete this and all other GSP assignments:
•
Although you will be emailing your assignment to me, save a copy of all of your sketches to your disk.
•
Answer all of the questions posed to you in a text box at the bottom of the sketch you are currently working
on.
•
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.”
1. In class we discussed how to construct an equilateral triangle (rather than just draw one.) In this sketch you
will learn how to construct a square rather than just draw one. (That is, you will learn how to construct a
square in such a way that the property of being square is built into the construction.)
Draw a line segment near the center of your sketch and label the endpoints A and B. Select the point A and
the segment
. In the Construct menu, choose Perpendicular Line. This creates a line through the point
A perpendicular to the segment
. Label the line you just constructed L1.
Now, select the point A and the segment
. In the Construct menu, choose Circle by Center and Radius.
This creates a circle with center A and radius
. Construct a point D at the intersection of the circle and
the line L1 above your segment. Now, hide the line L1 and construct a line segment connecting the points A
and D.
QUESTION 1: How do you know that the segment
Hide the circle that you used to construct segment
and congruent to
is congruent to the segment
. Construct a segment
that is perpendicular to
using a procedure similar to the one you used to construct
in your construction so that all that remain are the segments
,
, and
?
. Hide any objects used
:
Now, connect the points C and D with a segment (to form a square). Measure the four angles and the lengths
four sides of the square using the Angle and Length commands in the Measure menu. These measurements
should confirm that the quadrilateral you have constructed is a square.
QUESTION 2: How do your measurements confirm that ABCD is a square ?
Try dragging any of the vertices or sides of the square ABCD. You’ll notice that ABCD remains a square no
matter what is dragged since you have built that property into the construction. Color in the interior of the
square any color you like by selecting all four vertices in order around the square and choosing
Quadrilateral Interior in the Construct menu.
Save your sketch as GSP 2A (square).
2. In a new sketch, draw a generic quadrilateral ABCD. Construct the midpoint of segment
using the
Midpoint command in the Construct menu. Call this midpoint K. Similarly, construct midpoints L, M, and
N of the other three sides of the quadrilateral. Connect the midpoints to form a quadrilateral KLMN and
construct the interior of the quadrilateral KLMN as shown:
The quadrilateral KLMN is called the medial quadrilateral of quadrilateral ABCD.
You may have noticed that the medial quadrilateral appears to be a parallelogram, no matter how the
quadrilateral ABCD is shaped. (Drag points A, B, C, and D to see that this appears to be true.) This
observation is an example of a conjecture. A conjecture is a statement that seems to be true based on certain
observations. Formally stated, our conjecture would be:
The medial quadrilateral of any quadrilateral is a parallelogram.
Confirm (note: In this context the word “confirm” means to provide evidence in support of the conjecture; it
does not mean to provide a deductive proof of the conjecture) or reject the conjecture by making appropriate
measurements and observing how these measurements change as you change the quadrilateral ABCD. (Hint:
measure the slopes of the four sides of the medial quadrilateral using the Slope command in the Measure
menu. Don’t be alarmed if a coordinate grid appears on the screen– this happens anytime you measure a
slope).
QUESTION 1: Explain how your measurements confirm or reject the conjecture.
Save your sketch as GSP 2B (medial quad).
3. In a new sketch, construct a circle. Construct a diameter of the circle (i.e. a segment that goes through the
center of the circle with its endpoints on the circle.) (Hint: To construct a diameter, first construct a line– not
a segment– through the center of the circle to find the endpoints of a diameter, and then construct a segment
connecting those endpoints.)
Construct a point (label it P) on the circle. Construct a triangle by connecting point P to
the endpoints of the diameter (see figure):
Observe the triangle as you drag the point P around the circle.
Make a conjecture about the type of triangle that you have constructed. Try to confirm
or reject your conjecture by making appropriate measurements. If your measurements reject your conjecture,
modify your conjecture. Repeat until you have a conjecture that you are confident is true.
QUESTION 1: State your conjecture.
QUESTION 2: Explain how the measurements you made confirm your conjecture.
Save your sketch as GSP 2C (diameter triangle).
4. Do problem #35 on page 636 of your text. Be sure to answer the questions posed to you in parts (a) and (b)
of the problem in a text box.
Save your sketch as GSP 2D (p. 636 # 35).
5. In a new sketch, construct a regular hexagon ABCDEF (see the figure below for an idea of how to accomplish
this).
Hide all of the circles that you used in your construction so that all that can be seen are the sides, the vertices,
and the center of the hexagon. Label the center of the hexagon G.
Confirm that the Angle Measure in a Regular n-gon theorem (found on p. 627 of the text) holds for your
hexagon by measuring an interior, central, and exterior angle of your hexagon. (Hint: To measure an exterior
angle, you will need to construct a ray and a point on that ray. For example, to create an exterior angle at D,
you could construct a ray that originates at the point E and goes through the point D. Then, to measure that
angle, you would need to put a point on the ray out beyond D).
QUESTION 1: Explain how your measurements confirm the theorem.
Save your sketch as GSP 2E (regular hexagon).