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 ___ and ____ are of the same cube
  3. With these 2 facts, try to visualize what the blocks look like.
  4. The total number of cubes is ____ blocks

  5. The rectangle has a total of _____ cubes.

  6. Subtract the number of cubes from the total to get the number to fill it out = __________
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 total areas of the 2 triangles with 2 unit heights:


    A = _______ sq. units
  2. The 1 triangle with 1 unit height:

    A = sq. units.
  3. Subtract the area of the triangle with the 1 unit height from the combined areas of the larger 2 triangles = _______ 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   _____.
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) 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 sum of the other 2 angles of the triangle = _______.

  2. A ratio of 5:6 means there are _____ total elements of which the first is _____ and the second is _____ of the remaining angle.
  3. Multiply these fractions by the total of the measure of the 2 angles to get each angle =


    _____ and _____
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 = _____ feet.

  2. The diameter, d = ______ feet



  3. The height is ____________ feet