Pre-Lab: Area of trapezoid from length of middle segment.

Class and Lab Activity: More about strips and area. Link to JSP figure. (Need to use Netscape or Mozilla for this and probably not Explorer.)

Math 487 Lab 4: Carpenter and Right Triangle Lore

Carpenter's Construction

1. Work through Exp 4.1, Investigation 1: Carpenter's Construction with Pencil and Paper

In addition to the dots, if the corner is C, you may find it helpful to draw the whole angle ACB for a bunch (that's a technical geometry term) of locations for C.

2. Continue the same Investigation 2: Simulation of Carpenter's Construction with GSP

You should understand why the Sketchpad trace is doing the same thing that the did with the card.

Important: Make a Tool called Right Angle Locus. Once you are sure what object the trace turns out to be, make a geometry construction that starts with A and B and ends up with this object. Then make a tool that will carry out this construction.

Folding Right Triangles

This is pursuing the same set of ideas as the previous section. You should be looking for the connection.

3. Work through Exp 4.2, Investigation 1: Paper-folding experiment 

4. Work through Exp 4.2, Investigation 2: Folding Right Triangles with GSP

This is your first encounter with Line Reflection. To reflect an object across a line, first select the line and choose Transform > Mark Mirror. Now the mirror is marked (for keeps or until you mark the next mirror) so you can unselect the line. Then select what you want to reflect and choose Transform > Reflect.

Important: If A' is the reflection of A across line m, how is m related to segment AA'?

The goal of this section is to suggest reasons why the set of right angles is what you found in the first section.

Construction Solutions Using the Carpenter's Ideas

5. You can use your Right Angle Locus tool whenever you need to find points C so that ACB is a right angle. Here is an example:

• In a new figure, draw a line m and two points A and B not on the line. Construct all the points C on m so that ACB is a right angle.
• Solution: Use your Right Angle Locus tool to construct all possible C in the plane. Intersect this object with m to solve the problem.

6. Please read these instructions carefully.  Carry out the Construction of the first investigation in Exp 4.4, but wait before working through the Experiment.  Carry out these steps instead of the Experiment in the text.

   ·        Drag the point C so that line BC is tangent to the circle.  Where is point D when this happens? 

Question 1.  What property of tangent lines and radii causes D to be the point of tangency?

·        Trace the point D as before as you drag C around the plane.  The trace should look like a circle.  Figure out the center O of this circle and the radius.  You should be able to urse your Right Angle Locus Tool to construct the circle.

Question 2.  What is special about line BC when D coincides with one of the points of intersection of the two circles?

·        Intersect the two circles to get point F and G.  Construct lines BF and BG. 

Question 3. What is special about these lines? What is an explanation of why this construction works?

7. Ignore the rest of this investigation and move on to Exp 4.4, Construction Problem 1, Tangent Construction Script.

Make a Custom Tool called External Tangents using the construction given above.

On paper with straightedge and compass, construct the tangents to the circle through P.

8. Solve Construction Problem 2 of Exp 4.4 with Sketchpad.

Note how you would do this on paper with straightedge and compass also.

Conclusions about right triangles:

Be able to prove the following statements.

9.      Suppose ABC is a right triangle with right angle C, with D, E, F the midpoints of sides BC, CA, AB; prove the following:

·        Triangle AFE is similar to ABC.

·        Line FE is perpendicular to AC.

·        FE is the perpendicular bisector of AC.

·        FD is the perpendicular bisector of BC.

·        F is the circumcenter of ABC.

·        Triangles AFE, CFE, FBD, FCD are all congruent.

Conclusions about circles and the Carpenter's Theorem:

These proofs will be in Assignment  5B: (due Monday)

Prove:

If ABC is a right triangle, with right angle C, prove that the circumcenter of the triangle is the midpoint of the hypotenuse.

Or, in other words, prove that C is on the circle with diameter AB. (Why is this the same?)

Prove: 

If AB is a diameter of a circle, and C is a point on the circle, prove that angle ACB is a right angle. 

Hint:  Let O be the center of the circle.  Segment OC divides triangle ABC into 2 isosceles triangles.  Look at all the angles and their sums.