Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 7 Geometry
Problem
Hint
1) A wooden rectangular block measuring 9 inches by 12 inches by 16 inches is painted blue. The block is then cut up into 1-inch cubes. How many of the 1-inch cubes have blue on exactly one side?
  1. The cubes that have blue on only one side are the ones on the outside that are not the edges or corners.
  2. By subtracting 2 off each dimension of each side we can compute the number of cubes on each side that have only one blue side.



  3. The total number of blocks with one side painted is _____
2) In right triangle ABC, the measure of angle 1 is 40% of the measure of angle 2. What is the measure of angle 3?
  • Angles 2 and 3 add to 180 degrees so
    angle 2 = ______
  • Angle 1 is ______



  • Compute angle 3 from this information = _____

Problem
Hint
3) Mrs. Frieze is designing a geometric border (see figure below)to go around the top of her classroom wall. It's a repeating design and you can see a part of it in the diagram. The indicated angles at P are congruent and the indicated angles at Q are congruent. The angles at A and B are 90 degree angles. What is the length of QC?
  1. The angle at point D is equal to the angles at point Q
  2. This makes triangles BPQ and ADP similar.
  3. Side X on the diagram is the same length as side QC.
  4. Compute the length of side X from this information = _____
4) Demi is charged with finding the dimensions of a can that will hold twice as much as the 9 by 12 cm can as shown but have the same proportions. Her first step is to try a simpler problem. She computes the dimensions of the rectangle that would have twice the area but the same proportions as the shaded rectangle shown with base 9 cm and height 12 cm. What is the ratio between the height of the bigger rectangle to 12?
  1. The area of the rectangle is ______ sq. cm.
  2. The height of the rectangle is ____ of the base length
  3. The larger rectangle area = ______sq. cm.
  4. Let B = the new base length, H = new height


  5. Solve for B = ____ and H ______
  6. Ratio of the heights = _______
5) Triangle RST is the image of triangle ABC under a reflection and a translation. What is the translation? Describe it in terms of units moved up or down and units moved left or right.
  1. You must first do the reflection before you can do the translation.
  2. Reflect triangle ABC around the x axis to produce the reflected image and draw it on the image. To reflect around the x-axis means to change (x,y) to (x,-y).
    The new points are
    A' = (___,___), B' = (___,___),
    C' = (___,___)
  3. Now figure out what translations are necessary to match triangle RST = ____________________