Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 8 Geometry
Problem
Solution
1) The area of a square is the same as the area of a rectangle. The rectangle has a width of 6 units. The side of the square has length S. The rectangle has length L and is 4 times as long as S. What is the area of the square in square units?
  1. L = 4S
  2. Set the areas of the square and the rectangle equal to each other:
        S2 = 4S x 6 = 24S
        S = 24
        S2 = 576 sq. units.
2) Write the letters M, T, and H on figure 2 so that when the figure is folded into a box, it will spell MATH around the sides of the box. Figure 1 shows the correct placement of the letters
  1. Obviously the M goes in the square to the left of the A
  2. The spaces above and below the A cannot be where the T and H go (they fold above and below the A), so they go in the lower right squares.
  3. The T goes in the upper right square and the H goes in the lower right square, but with what orientation?
  4. Picture in your mind that T folding up and then around to be next to the A. The T should be as shown in Figure 2, to the left
  5. The H goes through 2 folds, first with the T and the second by itself to put it in the desired position. The H should be as shown in figure 2, also.
3) The hexagon shown has line of symmetry BE. The measure of angle FAB is 150 degrees, the measure of angle ABC is 114 degrees and the measure of angle DEF is 100 degrees. What is the measure of angle CDE?
  1. The sum of the angles of a polygon is 180(n - 2) where n is the number of sides (and angles) of the polygon.
  2. For our hexagon, that is 180 x 4 = 720 degrees.
  3. Because of the line of symmetry, we know 4 of those angles:
    FAB = BCD = 150 degrees
    ABC = 114 degrees
    DEF = 100 degrees
  4. The other 2 angles are equal, again because of the symmetry, so they are:
    720 - 2x150 - 114 - 100 = 206 degrees
  5. Angle CDE and FAB are equal to 206 so, CDE half of that = 103 degrees

Problem
Solution
4) There is a translation and a rotation taking vertices E, F, and G to R, S, and T. Where does vertex H go under this translation and rotation? Give your answer as a coordinate pair.
  1. Rotate EFGH around the origin 180 degrees (either way): This rotation translates (x,y) to (-x,-y), so the rotated coordinates of EFGH are (-3,+2), (-3,+1), (-7,+1) and (-5,+3)
  2. Then translate it -1 in the X direction. The coordinates of H under this rotation and translation is (-6,+3)
5) The latitude of a point, P, on the earth's surface is the measure of angle PCE, where C is the center of the earth and E is the point on the equator closest to P, see Figure 1. One night Joey measures the angle between a vertical pole at P and the direction of the North Star to be 50 degrees and plots this on the diagram in Figure 2. If the line from the center of the earth through the North Pole to the North Star is parallel to line of sight from P to the North Star, then what is the latitude of point P?
 Because the lines to the North Star from both the center of the earth (C) and P are parallel, the angle at P (50 degrees) is complementary with angle PCE (they add to 90 degrees together), so angle PCE is 90 - 50 = 40 degrees