Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 8 Geometry
Problem
Hint
1) What is the maximum number of 3 by 5 by 2 blocks that can be fit into a 16 by 23 by 10 box? The blocks must be stacked with the same orientation.
  1. Find the 2 dimensions of the blocks that divide evenly into dimensions of the box:


  2. The 3rd block dimension, ____ divides into ____ = ____ times (with a remainder)
  3. So multiply these factors together to get the number of blocks that divide into the box =


    ___________________

2) Triangle ABC has vertices at (2,3), (1, -2), and (-3, 1). The triangle gets reflected across the y-axis, rotate 90 degrees around the origin (counterclockwise ↶ ), then translated down three units. What are the coordinates of the resulting image's vertices?
 Plot the 3 points on this grid.
  1. Reflect across the y-axis
    (this means multiplying y axis values by -1):
        (2,3) = (___,___)
        (1,-2) = (___,___)
        (-3,1) = (___,___)
  2. Rotate 90 degrees counterclockwise
    (this means exchanging x and y values and negating the y value):
        = (___,___)
        = (___,___)
        = (___,___)
  3. Translate 3 units down (subtract 3 from the y value)
        = (___,___)
        = (___,___)
        = (___,___)

Problem
Hint

3) What is the measure of angle CDB? Not drawn to scale
  1. The angle at B inside ACDB is supplementary with the 121 degree angle and is therefore

    _______ degrees
  2. The angle at A inside ACDB is vertical with the 85 degree angle and is ____ degrees.
  3. So, using the fact that the angles of a quadrilateral add to ______ degrees, the angle CDB =


    _________ degrees.
4) Give the length of side RS in radical form.
 Using the pythagorean theorem determine the length of segment aR. Work to the left, computing the length of each segment up to RS. You should notice a pattern.






Segment RS = ______________
5) ABCD is a square. A, B, C, and D are the centers of their circles. The two smaller circles are congruent, and the two larger circles are congruent. DE = 4 inches and FA = 2 inches. Find the area of the shaded region. Express the answer to the nearest tenth.
  1. The area of ABCD = ______ sq. in.
  2. The 2 circular quarter circles centered at D and B together constitute a half circle of radius ____ inches, so their combined area is:

    _____________ sq. in.
  3. The 2 circular quarter circles centered at A and C together constitute a half circle of radius ____ inches, so their combined area is:

    ______________ sq. in.
  4. The area of the shaded area is the square's area minus these 2 half-circle areas:

    = _____________________ sq. in.