Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 8 Geometry
Problem
Solution
1) The parallel lines are 1 unit apart. What is the area of the shaded pentagon?
There are 3 triangles to consider: the 2 that have the 4 unit base and 2 unit height and the 1 that has a 1 unit height and 4 units base.
  1. The 2 triangles with 2 unit heights:
    A = 4x2 = 8 sq. units
  2. The 1 triangle with 1 unit height:
    A = 4x1/2 = 2 sq. units.
  3. Total area = 8-2 = 6 sq. units.
2) Mr. Bill is going to make a pouf (a foot rest) out of a rectangular piece of leather that measures 6 feet by 3 feet. His pouf will be a cylinder and he will stuff it with newspapers. The top and bottom will be circles, and the side will be made from one long rectangle that wraps around the top and bottom circle, see the shaded region. He will use the layout below that has the height of the pouf smaller than the diameter. How tall will he make his pouf in feet? Give your answer in exact form in terms of or as a decimal to the nearest tenth of a foot.
  1. The width of the piece of leather is the circumference of the pouf = 6 feet.
  2. The diameter, d = 6/
  3. The height is 3 feet - the diameter =
    3 - 6/
3) Anastazia is climbing a very steep hill by a path that goes due East. It turns out that the distance she has traveled East (as the crow flies) is the same as her gain in altitude. Using her GPS receiver she calculates that she has gone 2000 feet directly East. How far has she actually walked straight up the hill? Give an exact answer. The figure to the left shows Anastazia's path. By the pythagorean theorem:

D = √ 2000 2 + 2000 2  =
√8000000 = 2000√2

Problem
Solution
4) A cube sculpture is made from cubes glued together along their faces. Look at the three views provided of Joe's cube sculpture. After he put it together he painted all exposed surfaces. How many square faces of the unit cubes did he paint?
  1. The figure to the right is what this sculpture looks like.
  2. Strangely, you don't even need to visualize the 3-D structure to count the painted faces because they are twice the exposed faces! (one you see and the other one on the back).
  3. So,
    Top view: 5x2 = 10 faces
    Front view: 5x2 = 10 faces
    Right Side view: 6x2 = 12 faces
  4. Total = 10 + 10 + 12 = 32 painted faces
5) The triangle ABC shown has the angle measures shown. It is reflected over the line BC. Then the image triangle is reflected over the image of line BA. Keep reflecting the new image triangle over new line segments that have one endpoint at B until you have 12 triangles total all with one vertex at B. Is the resulting shape a polygon and if so how many sides does it have? If it's not a polygon why isn't it? A sketch can be part of your answer.
  1. The angle ABC = 180 - 80 - 70 = 30 degrees
  2. Reflecting the triangle around line BC and then the reflected triangle around the reflected line BA results in a folding around a circle.
  3. With 12 triangles (the original one +11 reflections) makes the total angle at B = 12 x 30 = 360 degrees.
  4. The resulting figure closes on itself and it is a polygon. It is a dodecagon (see the figure to the right)