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CTSE 7970, Summer 2001, Lab 5 — Geometry’s Sketchpad II

Section I — Explorations

1.   Midpoints!

a.   Draw a triangle. Then form a second triangle formed by joining the midpoints of its sides.

b.   Make several conjectures about how the second triangle compares to the original. Prove your finding(s)?

c.   Now drawn ABCD, an arbitrary quadrilateral. Construct E, F, G, H, midpoints of the corresponding sides. What can you find about quadrilateral EFGH. Prove your finding(s).

d.   What do you think should happen with a pentagon? Draw a sketch to see if your predictions are right. Then prove your finding(s).

2.   Centers of triangles.

a.   Construct the three perpendicular bisectors of a triangle. What can you find? Does your finding hold for other shapes of triangles. Define the CIRCUMCENTER (C) of a triangle. Prove the finding.

b.   Construct the three angle bisectors of a triangle. What can you find? Does your finding hold for other shapes of triangles. Define the INCENTER (I) of a triangle. Prove the finding.

c.   Construct the three altitudes of a triangle. What can you find? Does your finding hold for other shapes of triangles. Define the ORTHOCENTER (H) of a triangle. Prove the finding.

d.   Construct the three medians of a triangle. What can you find? Does your finding hold for other shapes of triangles. Define the CENTROID (G) of a triangle. Prove the finding.

e.   Construct  G, H, C, and I in the same triangle.  What relationships can you find among G, H, C, and I or subsets of them?  Explore for many shapes of triangles.

3.   In the diagram below, find the shortest path for the boy to go to the river to let his horse drink some water, and then go home. Find a general result.

4.   AB is a diameter of circle O. The radius OD is perpendicular to AB. M is the midpoint of OD. Ray BM intersects the circle at C. Find the length of BC.

Section II — More constructions

5.   We’ll start with a few basic constructions. Many of these constructions will require you to copy a length, so you need to draw a circle with a given radius. You might find the command useful: “Circle with Given Center and Radius”, under the Construct menu.

a.   Construct a segment equal to a given segment.

b.   Construct the perpendicular bisector of a given segment.

c.   Construct an angle equal to a given angle.

d.   Construct the bisector of a given angle.

e.   Construct a triangle having its three sides respectively equal to three given line segments.

f.    Construct a triangle having two sides and the included angle equal to two given line segments and a given angle.

g.   Construct a triangle having a side and two adjoining angles respectively equal to a given line segment and two given angles.

6.   Here are a few that are more challenging:

a.   a triangle, given the base, altitude, and one of the other sides.

b.   a triangle, given the base, altitude, and an angle at the base.

c.   a right triangle, given a leg and the altitude on the hypotenuse.

d.   a right triangle, given the altitude on the hypotenuse and one acute angle.

e.   an isosceles triangle, given the base and the altitude on one of the equal sides.

f.    an isosceles triangle, given the altitude on the base and a base angle.

g.   a right triangle, given the sum of the legs and an acute angle.

h.   a triangle, given the three medians.

i.    a triangle, given two angles and the perimeter.

7.   And now some quadrilaterals:

a.   Construct a square, given:

b.   Construct a rectangle, given

c.   Construct a rhombus, given

d.   Construct a parallelogram, given

Section III — More explorations

8.   Let Triangle ABC be an arbitrary triangle in the plane, and let triangles A’BC, AB’C, ABC’ be equilateral triangles attached to the outside of triangle ABC. Explore this situation and try to find as much as you can. What do you think about this situation or these findings?

9.   STREET PARKING. You are on the planning commission for Algebraville, and plans are being made for the downtown shopping district revitalization. The streets are 60 feet wide, and an allowance must be made for both on-street parking and two-way traffic. Fifteen feet of roadway is needed for each lane of traffic. Parking spaces are to be 16 feet long and 10 feet wide, including the lines. You job is to determine which method of parking--parallel or angle--will allow the most room for the parking of cars and still allow a two-way traffic flow. (You may design parking for one city block (0.1 mile) and use that design for the entire shopping district.)

10.   Triangle ABC is an arbitrary triangle. BD = (1/3)BC, CE = (1/3)CA, and AF = (1/3)AB. Triangle PQR is formed by the construction of line segments AD, BE, and CF. What is the relationship between triangle PQR and triangle ABC?

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