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

 

Do this construction on the Report 4 Sheet and draw a complete circle.

 

Answer in Writing: Q1

 

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

 

Answer in Writing: Q1, Q2

Folding Right Triangles

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

 

Answer in Writing: Q1, Q2

 

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

 

Answer in Writing: Q1, Q2, Q3, Q4, Q5

 

Construction Solutions Using the Carpenter's Ideas

 

  1. 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 if this circle.  Construct O in your figure and construct the circle.  (You can turn off the tracing of D now.).  So now you should have a real circle and D should appear to travel on the circle as you drag C.  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?

 

Answer in writing, Q1, Q2, Q3 above

 

  1. 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.

 

  1. Solve Construction Problem 2 with Sketchpad.

 

Given the lengths a and c on the Report Page, construct with straightedge and compass a right triangle ABC with right angle at C, AB = c and BC = a.

Conclusions about right triangles:

Write the proofs of the items below in Lab Report 4:

 

8.      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:

Write the proofs of the items below in Assignment  4: (due Monday, not the Lab Report)

 

  1. Prove:

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

Hint: Use what you proved in the previous section.

 

  1. 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.