Lab 5: Wed 10/31/01

This lab will be taken from GTC Chapter 5, BUT SOME SECTIONS WILL BE SKIPPED AND OTHERS WILL BE TAKEN OUT OF ORDER OR MODIFIED. So read the directions carefully.

Part A. Tangent lines and perpendiculars

Definition: Given a circle c, a line m is tangent to c is m intersects c in exactly one point.

The goal of this part is to quickly see the ideas behind the following Theorem.

Theorem. Let c be a circle with center C. For any point A on c, a line m through A is tangent to c if and only if m is perpendicular to CA.

Notice that this is an if-and-only-if statement, so there are 2 things to prove. Write down separately what these two if-then statement are.

Go to Exploration 5.2, Investigation 2. Carry out the construction of the distance slider in the first figure and also add the perpendicular CE as in the second figure. Do not do all the experiments or answer the questions. Instead, answer these questions.

Explain why |CD| is always greater than |CE| if D is distinct from E (what theorem do you use?).

Now construct the circle with center C through D. There appears to always be a second intersection point D’ of the circle with line AB. Explain how to prove there is always such a D’ by telling how to construct point D’ before you draw the circle. (Hint: How is line CE related to segment DD’?).

Put these two explanations together to prove the Theorem.

Part B. Circles tangent to two lines.

Carry out Investigation 3 of Exploration 5.2. BUT WITH ONE MODIFICATION.

Do the first construction in the investigation.
Then construct the angle bisector of FHG (not AHD yet). Then drag E around so that the two circles coincide, but do this with E in each of the 4 wedge-shaped regions (angle regions) created by the two lines.
Now construct the 4 angle bisectors of the 4 angles defined by the two lines.
What you are seeing is this fact (which you will prove in the homework). The locus of centers of circles tangent to two lines m and n intersecting at O consists of the two perpendicular lines p and q (also intersecting at O) that bisect the angles defined by m and n (excluding point O itself from the locus).

Part C. Coins tangent to two lines.

Carry out Investigation 1 of Exploration 5.2. You can try it with actual coins if you like, but also do it with Sketchpad.

In addition to connecting the centers to form the sides of a quadrilateral, also construct the diagonals. How are the diagonals related to the two intersecting lines?

Part D. Some Practical Straightedge and Compass Constructions

Use the insight you have gathered so far to carry out these constructions with Sketchpad. The same constructions should work with straightedge and compass.

Construction 1. Construct rays AB and AC and construct the angle bisector of BAC. Given any point P on the angle bisector, construct a circle with center P tangent to AB and AC.

Construction 2. Construct rays AB and AC. Let D be a point on ray AB. Construct a circle tangent to both AB and AC whose point of tangency with AB is D.

Construction 3. Construct rays AB and AC. Also construct a segment DE; denote by d the length of segment DE. Construct a circle tangent to rays AB and AC which has radius length d.

Construction 4. Construct rays AB and AC. Also construct rays BA and BD. Construct a circle that is tangent to lines AB, AC, BD. (Hint: Where are the centers of circles tangent to AB and AC? Where are the centers of circles tangent to BA and BD?) Make a script that will take A, B, C, D and construct the circle.

Construction 5. (A special case of Construction 4.) Given a triangle ABC, construct a circle interior to ABC that is tangent to all 3 sides.

Part E. Strips and angle bisectors

Carry out the Construction on page 68– Constructing a Strip from the midline and a Length (NOT THE EARLIER ONES). Make a script as explained there.

Do Investigation 3 of Exploration 5.3, at least the Strips for Two Intersecting Lines part. Do the rest if time permits.

The Assignment Due Friday 11/2 is based on the ideas of this lab.