Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 8 Geometry
Problem
Hint
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.) Because of the rotational symmetry the 2 large triangles are equivalent as well as the 2 small ones.
Use the pythagorean theorem to determine the triangle side lengths and then assemble all the lengths.






Total perimeter = __________
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 = _____, and so is B. Use the pythagorean theorem to compute C:




C =

Problem
Hint
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 = ______ degrees

  2. The angle that is vertical with the 36 degree angle = _____ degrees.

  3. The third angle of that triangle is ______ degrees.

  4. Angle q is supplementary with this angle, so q = ____________
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:
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.
Notice that if you know angle TAC then you also know angle TAB because those 2 triangles have sides of the same length.
Together, with angle CAB, they add to _____ degrees.
From this information you should be able to find out how many measurements you need:






You need _____ measurements.