Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 8 Geometry
Problem
Solution
1) Farmer Judy is creating an oddly shaped rotationally symmetric raised garden bed. She needs to buy lumber to frame the bed. What is the perimeter of the bed? Express to the nearest foot? (The dashed lines are to indicate lengths and are not part of the framing.)
  1. Because of the rotational symmetry the 2 large triangles are equivalent as well as the 2 small ones.
    We will need the pythagorean theorem to determine the triangle side lengths.
  2. The side length S of the larger triangle is:
    S = √ 3 2 + 2.5 2)  = 3.9 ft.
  3. The side length s of the smaller triangle is:
    s = √ 1 2 + 2 2)  = 2.24 ft.
  4. Assemble all the lengths:
    2.24x4 + 3.9x4 + 4x2 = 4.48 + 7.9 + 8 = 32.56 = 33 ft.
2) The diagram is a net for a pyramid with a square bottom with highest point above a corner of the square base. The outside solid lines are cut lines, the interior dotted lines are fold lines. If the side length of the square is 3 units, and the height of the pyramid, which is also its vertical edge, is 4 units, then what is the length of its longest edge C? Give an exact answer or express to the nearest hundredth. When folded, sides A and B (see figure) will be next to each other. Side A is a 3-4-5 triangle, so A = 5, and so is B. Using the pythagorean theorem we compute C:
C = √ 5 2 + 3 2)  = √34 or 5.83 ft
3) A section of a map looks like the figure indicates. The roads are straight but meet at odd angles. Two of the angles are given. What is the measure of the angle labeled q?
  1. The angle that is supplementary with the 54 degree angle = 180 - 54 = 126 degrees
  2. The angle that is vertical with the 36 degree angle = 36 degrees.
  3. The third angle of that triangle is 180 - 126 - 36 = 18 degrees.
  4. Angle q is supplementary with this angle, so q = 180 - 18 = 162 degrees

Problem
Solution
4) A pinwheel design is created by rotating the triangle shown about the origin to create 4 equally spaced copies each one having its interior in a different quadrant. The vertices all have integer coordinates. What are the coordinates of the copy of the triangle with interior in the fourth quadrant?
  1. To get the triangle into the 4th quadrant requires a 90 degree clockwise (↷) rotation.
  2. To rotate (x,y) 90 degrees clockwise = (y,-x). Perform this transformation for each triangle vertex:
  3. (0,0) = (0,0)
  4. (1,0) = (0,-1)
  5. (2,3) = (3,-2)
5) Jerry wants to recreate the drawing at right using a larger circle. A is the center of the circle and segment TC is congruent to segment TB. What is the fewest number of angles he needs to measure and still be able to compute the measure of all the angles in the drawing? "Congruent", in this case, means the segments TC and TB are the same length.
  1. If you know angle TAC then you also know angle TAB. They are the same angle because they form isosceles triangles.
  2. Once you know TAC, you can know both ATC and ACT because they form an isosceles triangle with angle TAC.
  3. You also know angles ATB and ABT because they are the same as angles ATC and ACT.
  4. The only other angle at the center of the circle, A, is angle CAB which can be computed because you know angles TAC and TAB, which, when added to CAB equal 360 degrees.
  5. We're getting close!
  6. Once you know angle CAB, you can know both angles ACB and ABC because they, too, make an isosceles triangle with angle CAB. That is All the angles of the figure so, the answer is
    1 measurement.
    All you need to know is angle TAC or TAB and all the other angles can be derived from them.