Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 6 Geometry
Problem
Hint
1) A cube has the same numeric value for its surface area and its volume. How many units is its side length? Use s for the length of the cube's side.
1. The surface area of a cube is 6 times the square of the side length.
2. The volume of the cube is the side length cubed.
3. Write an equation setting these 2 things equal to each other and solve for the side length s = _____

2) JIHG is a square with an area of 36 square inches.
JN is 1/3 of JI.
What is the area of square KNML?
1. Compute the side lengths of the large square = ____ in.
2. If KNML is a square, then all the triangles JNK, NIM, KGL and LHM are congruent (identical).
3. All the triangles have a short side JN = ____
    and a long side NI = _____
4. Find the area of one of the triangles = _____
5. Multiply it by 4 and subtract from the outer square area = _____
3). A parallelogram has vertices at
    (2, 3), (-6, 0), and (-2, 5).
Where is its 4th vertex located?		 The points are 1, 2 and 3 in the order listed.
Use this grid:
1. Plot the 3 points on the grid.
2. Determine the x and y offset of point 1 from point 3
    (this is going from 3 to 1) = (___,___)
3. Apply this offset to point 2 = (___,___).

Problem
Hint
4) Find the area of the figure
 1. Draw a horizontal line that touches 4 of the vertices and cuts the figure into 4 triangles.
2. Determine the base, height and area of the 4 triangles:
    1: (big one on the bottom) b = ____ h = ____ a = ____
    2: (left skinny one) b = ___ h = ____ a = ____
    3: (middle upper one) b = ____ h = ____ a = ____
    4: (right-most one) b = ____ h = ____ a = _____
3. Add these areas = _____ sq. cm.
5) Angle A has the same measure as angle B.
What is the measure of angle A?
 The figure is a pentagon.
Method 1: Use the sum of the interior angles of a polygon formula = ________________
  where N is the number of sides.
1. The sum of the angles of a pentagon is _____

2. Add the 3 given angles = _____

3. Subtract this from the pentagon angle sum = _____

4. Divide by 2 = _____

Method 2: Construct a quadrilateral
  1. Construct a line from point E to point C on the figure to the left:
  2. Angles DEC + DCE = _____ degrees
    (They make 2 angles of the right triangle DEC)

  3. Angles AEC + ECB = ________ degrees.

  4. For the quadrilateral ABCE, the angles are:
    ______ + 2A = 360 degrees.

  5. Therefore, A = _________
Editor's note: If you don't know the formula for the interior angles of a polygon you must use method #2)