Instructions
Do 5 or 6 problems, including at least one of #5 or #7. Problem 8 is worth extra. Problem 9 is worth extra extra.
Problem 1
Draw a segment AB and construct a point D on AB. (D should slide along the segment.) Construct (using the toolbar and Construct menu) a point C so that ABC is a right triangle with right angle ACB so that segment CD is an altitude.
Measure the lengths |AD|, |BD|, |CD|. Type into your sketch the statement of a relationship among these lengths and use the GSP calculator to verify it.
Save the sketch under the name prob1.gsp
Problem 2
Draw a circle with center A through B. Draw a point C outside the circle. Construct points S and T on the circle and lines CS and CT so that these lines are tangent to the circle.
Problem 3
Construct a segment AB and draw points C and D on the same side of line AB. Construct rays AC and BD. Construct a circle which is tangent to the segment and the two rays.
Problem 4
Construct an equilateral triangle ABC. Let A
a, etc., denote rotation with center A by angle a, etc., as usual. For a given point P, construct P’’ = A60 B120(P).The figure out what this product isometry T = A
60 B120 is and construct the relevant mirror lines, centers, etc. Write in a text box where these important points or lines are in relation to ABC.Then draw a point Q and define Q’ = T (Q) directly from the transform menu (do it as one transformation; it will have the same result as the composition, but do it in one step).
Problem 5
Do both these as separate figures in the same sketch. The given points should be free; the others should be as dynamic (movable) as possible.
Given two points A and B, construct points C and D so that ACBD (NOTE THE ORDER) is a rhombus.
Given two points M and N, construct points O and P so that MNOP (NOTE THE ORDER) is a rhombus.
Problem 6
Construct a parallelogram ABCD. Draw a point P on the plane. Construct points QRS so that ABCD is the midpoint quadrilateral of PQRS.
Problem 7
Construct a rectangle ABCD. Let G be the glide reflection with glide line AB and glide vector AB. Let H be the glide reflection with glide line BC and glide vector BC.
Investigate the produce HG and construct it as a GSP transformation.
Problem 8 (Extra points)
Given a segment AB, construct the regular pentagon with side AB.
Do not use the Transform menu for this problem.
Problem 9 (More Extra points)
Rose Window Construction. The stained glass windows of the cathedrals of the Middle Ages had beautiful frameworks that were constructed by geometry.
Here is an example of such a shape. Your task with Sketchpad is to begin with a circle c with center A through B. Then construct the arcs as in the figure. Neighboring arcs are tangent and each arc is tangent to c.
Hint: If you can find the ratio of the radius of C to the radii of the arcs, you can then construct the figure. On the paper, sketch in the center of the arcs (and the rest of the circles of which the arcs are a part). What kind of quadrilateral is formed by the centers? How is this figure related to the points of tangency of the arcs?
