Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 5 Geometry
Problem
Hint
1) The stained glass window design is a rhombus subdivided into four congruent rhombi. If the small angle at point M is 40o as shown then what is angle DAB?
 1. The other angle at M is supplementary to the 40o angle.
2. The other angle at point M is therefore _________ degrees.
3. Because all the shapes are rhombuses, all the opposite sides of the small diamonds are parallel, making the DAB angle the same as the above supplementary angle, making DAB = ______
2) A polygon has the two perpendicular lines of symmetry shown. One quarter of the interior of the polygon is the shaded trapezoid drawn in. Sketch the completed polygon and give its geometric name based on the number of sides or angles it has.
 How many sides does the full polygon have? _____
That shape is a __________.

Problem
Hint
3) Two congruent isosceles triangles are joined together along a side to create a kite as shown. The perimeter of one of the triangles is 30 units. The lengths of the sides of the triangle are all whole numbers. What is the length of the longest side of the triangle if the perimeter of the kite is as long as possible?
 Since the lengths of the sides are all whole numbers and the perimeter of each triangle is 30, you can make a table like this:      
PerimeterLong
side
(L)		Short
side
(S)		Kite
perimeter
2L + 2S
30101040
30    
30    
30    
30    
30    
Keep adding to the long side length until the short side length goes to zero. Pick the biggest side length that has a whole number short side.
4) Rosie builds a large cube out of sugar cubes. Her large cube has a width of 4 cubes, height of 4 cubes and length of 4 cubes. She wants to add one more layer of cubes all around to make an even larger cube that will have height, width and length 6 cubes. How many cubes does she need? 1. The number of cubes in her 4 x 4 x 4 large cube is ____
2. The number of cubes in a 6 x 6 x 6 large cube is ____
3. Subtract the two numbers.
5) For triangle ABC shown on the coordinate grid what are the coordinates of a 4th point (D) that turns the triangle into a parallelogram? 1. Since the desired figure is a parallelogram that means that side BD (with your new point D) is parallel to side CA.
2. Determine the offset in X and Y axes from point C to get to point A. Offset = (___,___)
3. Apply these offsets to point B to get point D = (___,___)