Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 7 Geometry
Problem
Solution
1) A cube sculpture is created from cubes stacked on top of each other and next to each other. Cubes that are stacked meet along their square faces. The top view, front view, and side view of a cube sculpture are given below, where only the squares directly facing you from each view point are shown. What is the least number of cubes you would need to add to the sculpture to make it a solid rectangular prism of cubes?
 In the figure to the left, the faces are numbered for reference.
  1. In the top view and the right side view faces 1 and 14 are of the same block because all others are in the front view.
  2. Faces 2 and 5 are of the same cube
  3. The total number of cubes is 6+1 = 7
  4. The rectangle has 3x3x2 = 18 total cubes.
  5. 18-7 = 11 cubes to fill it out
The figure to the right shows this figure.
2) 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.

Problem
Solution
3) Jill decided to rearrange her room. She will rotate the bed and table counter clockwise 90 degrees about the center of her room. The center of the room is given by the origin. The original coordinates of the table were (0, -4), (2, -4), (2, -6), (0, -6). What will be the new coordinates of the table?
"Counterclockwise" means this way: ↶
This problem is asking for the coordinates of the table, not the bed!

1. With a counterclockwise rotation, the table will be in quadrant I.
2. This will put it's edge on the x-axis in positive y coordinates.
3. Turn your head sideways to the right and try to visualize the new position of the table. Sketch it in on the graph.
4. The new coordinates of the table are :
    (4,0) (4,2) (6,0) (6,2)
See figure to the left.
4) One angle of a triangle has a measurement of 70 degrees. The ratio of the other two angles of the triangle is 5:6. What is the measure of each of the other two angles?
  1. The other 2 angles of the triangle sum to 180 - 70 = 110.
  2. A ratio of 5:6 means there are 11 total elements of which the first is 5/11ths and the second is 6 11ths of the remaining angle.
  3. 5⁄11 x 110 = 50 degrees.
  4. 6⁄11 x 110 = 60 degrees.
  5. The angles are 50 and 60 degrees
5) 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 in the picture 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/