MAT 2270: Problem Set 3
This problem set is designed to help you further explore the capabilities of Geometers Sketchpad. Complete
each problem and save your work for each as a new page within one Sketchpad file. Title each page with
a word describing the problem. Each question requires you to include a written response, submit this as a
separate document (not a Sketchpad document).
Be sure to label each response so that it corresponds with the problem it came from. Turn in your Sketchpad
file via e-mail before class on the day it is due. I will test each Sketchpad construction. Your written work
will be assessed for clear, accurate, and complete responses.
1. Construct right triangle XYZ that remains a right triangle under distortion. You must be able to
distort each legs length without changing the length of the other leg. Provide a step-by-step guide to
explain how the triangle was created. Include as part of your sketch a display of the degree measure
for each of the triangles interior angles.
2. Do the following:
(a) Construct rectangle ABCD. Be sure that the properties of a rectangle are maintained when the
original figure is distorted.
(b) Construct the diagonals of rectangle ABCD and label their intersection as P.
(c) Construct a new rectangle WXYZ that has vertices located at the midpoints of AP, BP, CP,
and DP.
(d) Repeat the process of steps (b) and (c) with respect to rectangle WXYZ. Label the new rectangle
EFGH.
(e) Repeat the process of steps (b) and (c) with respect to rectangle EFGH. Label the new rectangle
RQST.
(f) Display the area of all four rectangles.
(g) Display the ratio of the areas WXYZ:ABCD, EFGH:ABCD, and RQST:ABCD.
Identity the relationship between the areas of the smaller rectangles to rectangle ABCD. Does the
relationship hold as you distort the figure into rectangles with different dimensions? Suppose we
repeat this process to construct infinitely many new rectangles, how can you generalize the results
concerning the ratios of areas? What would be the result of the summation of these areas? Provide
appropriate justification for your responses.
3. Let triangle ABC be any triangle and let P be any point in the plane. Construct perpendiculars from
P to each side of triangle ABC. Label the intersection of each perpendicular with its appropriate
triangle side R, S, and T respectively. Triangle RST is called a pedal triangle for pedal point P.
Explore the following and make conjectures about the nature of triangle RST:
(a) What if P is on a side of triangle ABC?
(b) What if P is one of the vertices of triangle ABC?
(c) What if P is the centroid of triangle ABC? the incenter? the orthocenter? the circumcenter?
4. Construct a circle centered at the midpoint of the Euler segment and passing through the midpoint
of one of the sides of the triangle. This circle is called the nine-point circle. The midpoint it passes
through is one of the nine points. What are the other eight? Once you’ve determined the points,
drag the triangle around and investigate special triangles. Describe any triangles in which some of
the nine points coincide.
5. Construct a working windmill that contains four equally spaced blades. The windmill should be
operated by an action button. It should also rotate clockwise and have the property that no one
blade can be moved without the others also moving, or you cannot move the blades separate from
the action button. Provide details to explain your construction.
