Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 6 Geometry
Problem
Hint
1) The circumference of the circle shown is 6 units. It has center B. The shaded region has half the area of the unshaded region. What is the perimeter of the shaded region? Express your answer to the nearest tenth of a unit.
Use = 3.14 if needed. 		1. The formula for the circumference is
    C = D,
    so D must be _____
2. Segments AB and BC are radii, so their length is ____
3. Since the shaded region is half the unshaded region, then it is ____ of the circle and the portion of the circumference from A to C is ____
4. Add 2 of the radii and this AC circumference segment to get perimeter of the shaded region = _____

2) Line segments AB and CD are parallel. The three circles in between them just touch each line segment at one point and the circles touch each other at one point. Each circle has a diameter of 10 units. Find the area of the shaded areas in between the circles to the nearest square unit.
Use = 3.14 if needed.
1. The line segments AB and CD touch the circles at their half-way points.
2. You can make a rectangle whose corners are the points where the lines touch the outer circles on both the AB line and the CD line. Draw in that rectangle.
3. The idea here is to compute the area of this rectangle and then subtract the areas of the included circles. There are _____ circles included in this rectangle.
4. Since the diameter of each circle is 10 units, compute distance on the AB line from the first 'touch' point to the last one = _____
5. Using this and the diameter of the circles, compute the area of the rectangle = _____.
6. Subtract the areas of the included circles to get the area of the shaded portion = _____

Problem
Hint
3) Each figure below is made up of 1 square and 4 congruent isosceles triangles. The interior dotted lines are fold lines. Which of these figures can be folded to make pyramids with a square base? Here is a pyramid:
Study it and imagine each of the 4 figures folded into this shape. The failed shapes have sides that overlap when folded.

Hint: more than one of the shapes can be folded into a pyramid, but not all.
The figures that can be folded into pyramids are ____________
4) The coordinates of the vertices of a quadrilateral in clockwise order are: (0, 3), (4,8), (8,3), and (4, 5). What is its area?
Plot the points on the grid and number them.
Method 1: Cut into triangles and add:
1. Notice that the 2nd and 4th points have the same x coordinate = _____.
2. Cut the quadrilateral into two triangles along this vertical y-line
    (4,8) to (4,5)
3. Compute the areas of each triangle and add = ______

Method 2: Make a large triangle and subtract the missing triangle:
1. Compute the area of the triangle made by points 1, 2 and 3 = _____
2. Compute the area of the triangle made by points 1, 4 and 3 = _____
3. Subtract the second from the first to get the quadrilateral area = _____
5) The length of the shortest trip from A to B along the edges of the cube shown is 4 edges. How many different 4-edge trips are there from A to B? 1. The number of paths from point A are ____.
2. From that point (assuming you don't go backward
    or away from B) there are ____ paths forward.
3. From that point there are _____ paths to get to B.
4. From that point there are _____ paths to get to B and you are there!
5. Multiply all these paths together
    to get the total number of possible paths = _____