Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 5 Geometry
Problem
Hint
1) A parallelogram has vertices at (0, 0), (5, 2), and (1, 3).
The fourth vertex has positive coordinates. What are they? 		1. Plot the points on the graph.
2. Since the opposite sides of a parallelogram are parallel, the third vertex (1,3) and the 4th vertex are parallel to the first and second.
3. The offset of the second vertex from the first is
    going from the first to the second: (___,___).
4. Apply this offset to the third vertex to find the 4th =
    (___,___).
2) Jenny build a pyramid out of sugar cubes to model a Mayan pyramid. Each layer is a solid square of sugar cubes and the pyramid looks the same on all four sides. The bottom layer shows 10 cubes along each side. The next layer shows 9 cubes along each side and so on ending with one cube on top. Jenny paints all the surfaces of the pyramid that are showing. How many cubes have no paint on them? 1. If you lay out a square of cubes, 10 on a side (the bottom layer), then that makes _____ total cubes.
2. The cubes on just this one layer that will not have paint are the ones on the inside, making a square that is ____ cubes on a side.
3. Apply this to the remaining layers to compute the number of cubes that will not have any paint = _____ cubes.

3) At six o'clock the angle between the hour and minute hands on a clock is 180 degrees. What is the angle between the hour and minute hands on the clock at 6:30 pm? Assume that both hands move at a constant rate. 		1. At 6:30 the minute hand has moved 180 degrees from 0 to ____ degrees .
2. 30 minutes is _____ of a 12-hour clock face = ____ degrees. This is how much the hour hand has moved from 180 =
    ____ degrees.
3. Subtract the two to get the angle between them =
    _____ degrees.

Problem
Hint
4) Parallelogram ABCD has line of symmetry AC and has an angle of 120o at angle BAD. What kind of triangle is triangle ABC? 1. For parallelogram ABCD to have AC as a line of symmetry, triangle ABC must be congruent (equal) with triangle ACD.
2. Since the opposite angles in a parallelogram are equal, then
    angle BCD is also _____.
3. That makes the other 2 angles (ABC and ADC) both ______ degrees.
4. Line of symmetry AC cuts the angle at BAD in half, making angle BAC _____ degrees.
5. Since angle BAC and ABC are both _____ triangle ABC is ___________.

5) Two of the vertices of a parallelogram that contains trapezoid ABCD are (1,3) and (2,1). The parallelogram has twice the area of trapezoid ABCD, and its coordinates are positive. What are its other two vertices?
 1. If the parallelogram has twice the area of the trapezoid and contains points A and B and contains the trapezoid, then the remainder of the parallelogram must duplicate the trapezoid's area.
2. Draw 2 points on the grid that (a) complete the parallelogram, (b) contain the trapezoid and (c) doubles the trapezoid's area. The two points you add must be parallel to side AB.
    Points are (___,___) and (___,___).