Lab 3: Parallel Lines and Areas

This lab will be about parallel lines and areas. It has several parts.

Part 1. Arrangements of triangles

This is a way to make big triangles from little triangles. Follow this link to part 1.

Part 2. Midpoint Triangles and Shearing Triangles

For this part, return to the Getting Acquainted with Sketchpad from Lab 1. Work through Sections 1.5, 1.6, 1.7 if you have not already done so. WAIT a bit for Section 1.8 until later.

Part 3. Shearing Parallelograms

This is done very much like the shearing triangles from Part 2.

• Construct a segment AB and a line m parallel to AB through a point P. Hide P.
• Then construct a point C on m and then construct point D so that ABCD is a parallelogram.
• Again, construct a point E on m and then construct point F so that ABEF is a parallelogram.

Then we say that ABEF is constructed from ABCD by shearing. Both parallelograms have the same base segment and congruent heights, so they have the same area.

Part 4. Area formula for parallelograms in the coordinate plane

This section assumes some background information about the coordinate plane that we have not talked about in the course yet. But it is a terrific application of shearing of parallelograms.

Follow this link.

Part 5. Fun with Pythagoras

Follow this link for several sketches (already made) that illustrate some of the area proofs of the Pythagorean Theorem.