Work the following problems as assigned during the class. Write up your solutions and be prepared to discuss the problem in class the day they are due. If you cannot solve the problem completely, write up as much as you can, recording what you tried and what you needed to konw to be able to complete the problems. Problems similar to some of these will be on the final exam.
1. ABCD and EFGH are congruent squares with E the center of ABCD. Find the area of the intersection EJDK.
2. Some cubes are painted red on all sides. The first cube is cut once in each direction forming eight congruent cubes. The second is cut twice in each direction forming 27 congruent cubes. The third is cut three times in each direction, the fourth is cut four times, etc. If a cube is cut in this manner with n cuts in each direction, how many small cubes will be formed that have three red faces? Two red faces? One red face? No red faces?
3. Prove that from any point in the interior of an equilateral triangle, the sum of the distances to each of the three sides is a constant.
4. A checkerboard is of such size that one domino exactly covers two squares on the board. Thus, 32 dominos will cover the whole board. Now two squares are cut off the board, one at each end of the same diagonal. (Is it possible to cover the remaining 62 squares with 31 dominos? If yes, display a method; if no, prove it.)
5. Given an arbitrary triangle, inscribe a square in the triangle so that two vertices of the square lie on the base of the triangle and the other two vertices of the square lie on the two remaining sides of the triangle.
6. Find a point on the earth from which you can walk one mile south, one mile east, and one mile north and arrive back at your starting point. Now find three other distinct points with this same property.
7. Two sheep and a goat are chained at the midpoints of the sides of a right triangle. Each chain is just long enough to reach to the endpoints of that side, and the animals can graze in the resulting circular regions. How much grass is available only to the sheep, but not the goat?
8. In a class of 25 students, the desks are arranged in five rows of five desks each. The teacher directs the students to change their assigned seats so that each person sits in a different desk that is directly in front or directly behind or directly to the right or directly to the left of their original desk. Can the students carry out this directive? Prove your answer.
9. Four long segments are of equal length and four short segments are of equal length. The long segments are twice the length of the short ones. Use all eight segments and make exactly three congruent squares.
10. Inscribe an equilateral triangle inside of another equilateral triangle so that each side of the inscribed triangle is perpendicular to the corresponding side of the larger triangle. Find the ratios of the areas of the four regions into which the larger triangle is divided.
11. ABCD is a square with M the midpoint of DC. Find the ratios of the areas of the regions P, Q, R, and S.
12. Connect the pairs of boxes with like letters (R to R, B to B, and Y to Y) with simple curved lines. The lines must not intersect, and you must stay inside the rectangle.
13. In a room there are six nails driven -- one in the center of each wall, one in the center of the ceiling and one in the center of the floor. You have two colors of string with which to connect each nail to every other nail. Can you do this so that none of the triangles formed by the string have all three sides the same color. If yes, show how. If no, prove it.
14. The diameter of this circle is 10 units. A rectangle is inscribed in the circle, and a rhombus is inscribed in the rectangle. Find the length of the side of the rhombus.
15. Three houses are to be connected underground to water, electricity, and gas. For safety reasons, the lines are not to overlap. Can this be done?
16. All the ice in a cooler has melted. The cooler has rectangular faces and is sixteen inches high. When the cooler is tilted, the water just covers an end, but only three-fourths of the bottom. What is the depth of the water when the cooler is made level again?
17. What is the number of paths from A to B if one always travels east or north on the streets given?
18. A mathematician wants to play softball but needs to set the bases so they are at right angles to each other. A ten-foot tape measure and some string are the only available tools. How can the Pythagorean theorem be used to place the bases correctly?
19. A tree produces a single vertical stem one foot high during the first year of growth. The next year it produces from the tip of the original stem two branches each .5 feet long and at right angles to each other. The third year it grows two new branches from the tip of each of the previous ones, each .25 feet long and at right angles to each other. This growth pattern continues in successive years with each new growth half as long as the previous generation, and with each new pair of branches at right angles to each other. How high and how wide will the tree become?
20. ABCD is a square with M, N, O, P the midpoints of the sides. Find the area of the shaded region.
21. IJKL is a square with R, S, T, V the midpoints of the sides. Find the area of the shaded region.
22. A circle and a square have the same area. What is the ratio of the area of a circle inscribed in the square to the area of a square inscribed in the circle?
23. The sides of a trapezoid are one, two, three, and four centimeters long. What is the area of the trapezoid? Also, show that your answer is unique.
24. ABCD is a square with M and N the midpoints of AB and CD receptively. Find the area of the shaded region.
25. If the three vertices A, B, and C in the three adjacent squares are collinear, find the value of s.
26. If three congruent circles are tangent to each other as shown, and if the space between the circles is 100 cm2, find the radius of the circles.
27. How many rectangles are in the grid below? Generalize your answer to predict the number of rectangles in a grid of m by n units.
28. A snowflake begins as an equilateral triangle then three smaller equilateral triangles form, one at the center of each side of the first triangle. This process continues indefinitely as shown. What are the final area and perimeter of the snowflake?
Practice with Proofs
The following are some basic relationships. Give a formal proof of each one without the aid of books, notes, or other resources.
29. The sum of the interior angles of a triangle is 180 degrees.
30. The sum of the interior angles of any n-gon is (n 96 2)=95180 degrees.
31. The sum of the measures of the exterior angles of any polygon equals 360 degrees.
32. The line segment joining the midpoints of the legs of a trapezoid (called the median of the trapezoid) is equal to half the sum of the bases of the trapezoid.
33. The median of the trapezoid is parallel to both bases.
34. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length of the third side.
35. The segments joining the midpoints of consecutive sides of a quadrilateral form a parallelogram.
36. The segments joining the midpoints of the sides of a triangle divide the triangle into four congruent triangles.
37. The perpendicular segments drawn from the midpoints of two sides of a triangle to the third side are equal.
38. If one side of a triangle is greater than a second side, the angle opposite the first side is greater than the angle opposite the second.
39. Of all quadrilaterals of a given perimeter, find the one that has the greatest area.