Assignment 5 - Due Wednesday, 11/5

5.1 Regular Pentagon (10 points + 5 extra possible)

1. Draw a segment AB and construct neatly and accurately the regular pentagon ABCDE with straightedge and compass. (Make the segment at least two inches long. No microscopie figures please.)
2. Write a readable description of the major steps in your construction (e.g., construction of needed lengths, what points of C, D, E are constructed next and how).

   Important: If you have too many steps, write the description in outline form so that a reader can figure it out without close textual analysis. It is OK to be concise. Just say how the major features are constructed. You can do this with Sketchpad, but you still need to write the key steps.

5.2 Secant Lines (10 points)

Draw a circle c and a point P outside the circle. Draw a line m through P that intersects circle c in points A and B and draw a line n through P that intersects circle c in points C and D. (Such lines are called secant lines.)

• Find and prove a relationship among the 4 lengths PA, PB, PC, PD, using some similar triangles in the figure.

A plane of symmetry of a regular tetrahedron divides the polyhedron into two congruent parts.

5.3 Regular Tetrahedron

Part A. Make Models (10 points + 5 extra possible for accuracy and neatness)

1. Make a cardboard model of a regular tetrahedron ABCD from a net (plane pattern) of 4 equilateral triangles (see B&B, p. 188). The edge of the model should be at least a couple of inches.
2. A plane of symmetry will cut a regular tetrahedron into two congruent halves. Each half is also a tetrahedron, but not regular. Make an accurate net of 4 triangles for such a half of a regular tetrahedron. Then make a model of this half.
3. Bring the models and net to class..

Part B: Compute lengths and angles (15 points)

In all this, assume that ABCD is a regular tetrahedron wtih edge length AB = s.

1. On your net, one triangle is in the plane of the cut (the plane of symmetry). What are the side lengths of this triangle?
2. Find the height of the tetrahedron ABCD when it is resting on its base ABC. (Use your models for ideas.)
3. Give convincing reasons to show that there is a point P that is equidistant from the vertices, A, B, C, D. Find the distance from P to D. Show your reasoning.

Part C: Compute angles (10 points)

1. Tell how one measures the angle between two planes or two triangles with a common edge in space. (Such an angle is called a dihedral angle.)
2. Demonstrate your method by finding the dihedral angle between the faces of the regular tetrahedron. Your answer should be in two forms, first an exact answer in the form of an inverse trigonometric function of an exact number; then second, find a decimal approximation of this angle in degrees.