Lab 6: Area and Dissections

Part 1. Some Sketchpad from GASP

In Lab 1, several sections of the GASP handout were skipped. Look over these activities, and if you have not done them or do not recall them, do them now.

Go to this link and do these Sketchpad constructions:

• page 13, 1.5 Midpoint triangle
• page 18, 1.6 Shearing triangles
• page 19, 1.7 Areas and altitudes

Part 2. An outline of simple area formulas

This outline is excerpted from an assignment for the TLP program for future middle school teachers. Read over it pretty quickly and then move on to Part 3 to work out the same figures in Sketchpad.

Read the TLP area assignment (excerpts). Do the constructions with Sketchpad, as specified below.

Part 3. Areas of some polygons with Sketchpad

For this lab we will assume the axioms for area at this link (U. of Georgia). The figures that illustrate this Part 3 can be found at the link for Part 2.

Area of a Rectangle

Area of a rectangle is defined to be the produce of the width times the length. This includes the square as a special case.

1. Suppose R is a rectangle with area = A. If R' is a similar rectangle obtained by scaling distances in R by 2, what is the area of R'? ____________
2. Suppose S is obtained by scaling R by factor k. If area of S = 2 * area of R, what is k? _____________

Area of a Parallelogram

• Construct a parallelogram ABCD with Sketchpad. Construct the lines b and d through B and D perpendicular to AB (and CD).
• Construct the rectangle whose vertices are the intersections of b and d with line AB and line CD.
• Construct and color polygon interiors that make clear that the parallelogram ABCD can be cut up (dissected) like a jig-saw puzzle and the pieces reassembled as a rectangle. Conclude that the area of ABCD is the same as the area of the rectangle.

1. What lengths would you measure as the base and the height of ABCD? _______________
2. Use these to state the usual formula for ABCD. _______________________
3. Move the vertices CD so far to the side that m and n no longer intersect segments AB or CD. How can you justify the formula in this case?
4. Continuing with the same parallelogram, ABCD, repeat the construction of perpendiculars, but this time construct b' and d' perpendicular to AD and BC. Measure the base and the height from this construction. Are they the same as before?
5. Does the formula still give the same area? Why?

Area of a Triangle by doubling

Given a triangle ABC in Sketchpad, show how to construct a parallelogram ABCD whose area is twice the area of the triangle.

1. Deduce the area formula for a triangle from the formula for a parallelogram.

Area of a trapezoid

Given a trapezoid ABCD, with sides AB and CD parallel, construct another trapezoid A'CBD' congruent to ABCD so that AD'A'D is a parallelogram with double the area of the trapezoid.

1. Deduce the area formula for a trapezoid from the formula for a parallelogram. _________
2. If we shrink the segment CD to a point, how is the formula related to that of a triangle? _______
3. If CD = AB, how is the formula related to the formula for a parallelogram? _________

Area of a triangle by dissecting

Find the figure in Part 2 that shows how to cut a triangle into pieces that can be assembled into a parallelogram. Do this with Sketchpad. Hint: You will need a transformation to move the pieces so that they pass the drag test.

Part 4. Ratios Areas inside a triangle

Go to this link on ratios and areas inside a triangle. Some of this will appear in Assignment 6C.