Lab 5

This lab has several parts.

Part A: Special Midpoint Quadrilaterals

Lab Activity: Produce Sketches with dynamic counteraxamples to the false assertions in Assignment 4C, Problem 4.4.

USE THIS theorem, which a special case of a general Thales figure (see lab later)

Midline Theorem: Let ABC be a triangle, with M = midpoint AB and N = midpoint BC. Then MN is parallel to AC and |MN| = (1/2)|AC|.

Proof: Triangle ABC is similar to MBN with scaling ratio 1/2 by SAS for similarity.

• Thus |MN| = (1/2)|AC|.
• Since angle BAC = angle BMN by corresponding parts, the lines MN and AC are parallel by equal angles to transversals.

Corollary. In a quadrilateral ABCD, if M and N are as above, and O is midpoint of CD and P is mispoint of DA, then MN and OP are both parallel to the diagonal AC with length |MN| = |OP| = (1/2)|AC|.

Corollary. The midpoint quadrilateral MNOP above is a parallelogram. (Reason: Two opposite sides parallel and same length.)

Part B. Construction Practice

Try a few from this Construction Practice.

Part C. Parallels, trapezoids and ratios: Intro

First, do this prelimiinary work, then follow the link at the end for more about this geometry.

• Construct segment AB and a line through C parallel to AB. Construct D as a point on the parallel line.
• Construct lines AC and BD and their intersection O.
• Construct lines AD and BC and their intersection I.

Select 3 points on a line and choose Measure>Ratio. Do this for the following sets of 3 points (in order)

OAC, OBD, IAD, IBC. How do these numbers compare to the ratio |AB|/|CD|.

Consrruct the line OI. Let P be the intersection with line AB and Q the intersection with line CD. How are P and Q related to the segments AB and CD.

Construct the circle c1 with center P through A and the circle c2 with center Q through C. Then construct the tangents to c1 through O. How do these tangents relate to c2? Also construct the tangents to c1 through I (you may have to move the segments apart to for this).

Go on to this lab sheet that has more information: Lab 5 Dilations