Do all problems, but notice that in Problems 1 and 2 there is a choice. Do only one. If you work on both, indicate clearly which one should be graded.
Prove one of these two theorems using any tools you like (except for what you are proving).
A. State and prove the Pythagorean Theorem
B. Prove the concurrence of the medians of a triangle
Solve one of these problems. Show your work.
a) (20 points) Suppose that ABC is an isosceles triangle with AB = AC = x and BC = y. Suppose that D is a point on BC so that all 3 triangles ABC, ABD, ACD are isosceles. Figure out which segments are equal and then deduce the ratio x/y. NOTE: Read this carefully. It is not the same triangle that was an assignment problem.
b) (15 points) Let ABCD be a rectangle, with M the midpoint of AB and N the midpoint of CD. Then MNDA is also a rectangle. Let u = |AB| and v = |BC|. Then if MNDA is similar to ABCD, what is the ratio u/v?
Suppose that ABCD is a pyramid with an equilateral triangle ABC as base and 3 congruent isosceles triangular faces: ABD, BCD, CAD (i.e., DA = DB = DC). Also suppose that the dihedral angle between ABC and DBC is 45 degrees.
Suppose the side length of the triangle ABC is |AB| = s. Answer these questions. Show your work.
a) What is the height of the pyramid? _____________________________________
b) What is the volume of the pyramid? _____________________________________
c) What are the lengths DA = DB = DC? _____________________________________
In the figure below, all the chords have equal length. Recall the notation for rotation with center Z by angle a is Za.
Answer each of these questions. You do not have to give reasons for you answers.
a) How many rotational symmetries does this figure have (including the identity)?
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b) How many line reflection symmetries does this figure have? ______________
c) What is the measure of angle P8P5P2?
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d) If L1 is line reflection in line OP1 and M9 is line reflection in line OQ9, tell precisely what isometry is the product M9L1. (State defining data or use clear notation.)
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e) Continuing with L1 and M9 and with L5 = line reflection in OP5, tell precisely what isometry is the product L1L5M9. (State defining data or use clear notation.)
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Write the answer to these questions. You do not have to justify or explain your answer.
1. Suppose that U and V are glide reflections. What kind of isometry can UV be? (List the names of the possible types among the 5 kinds of isometries.)
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2. Let ABC be a triangle. If M is the midpoint of AB and N is the midpoint of AC and P is the midpoint of MN, what are the barycentric coordinates of P with respect to ABC?
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3. Given the triangle ABC below, what isometry is the composition of half-turns T = HAHBHC? For your answer sketch the defining data of the isometry T and also tell in words precisely the location relative to ABC using some geometrical figures.
4. If A = (23, 17) and P = (x, y), what is the formula for HA(P)?
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5. What regular polyhedron has the most faces? Write its name and the number of faces, edges and vertices.
Name: __________________________________________ No. of faces _____________
No. of edges _____________No. of vertices ___________
The problem refers to the figure on the next page. Suppose this figure is continued by translations to cover the whole plane. Ignoring the fact that one of the shapes is a different color, answer the following questions about symmetries of this figure.
a) If there are line reflections that are symmetries of the (infinite) figure, draw the mirror lines that pass through labeled points.
b) List all labeled points that are centers of 4-fold symmetry (i.e., centers of 90-degree rotations that are symmetries of the pattern)
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c) List all labeled points that are centers of point symmetry that have not been already listed above (i.e., centers of 180-degree rotations that are symmetries of the pattern)
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d) Let F denote rotation by 270 degrees with center D6 and G denote rotation by 270 degrees with center F4.
· What kind of isometry is GF? _____________________
· Tell its defining data using labeled points if possible.
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e) Let H be rotation by 90 degrees with center D6 and let M be reflection in line E3E6.
· Circle and label as "IMAGE" the image of the black shape by MH.
· What kind of isometry is MH? _____________________
· Tell its defining data using labeled points if possible. _____________________
