Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 7 Geometry
Problem
Solution
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. Here are the sides:
    • 2 x (9-2) x (12-2) = 140
    • 2 x (9-2) x (16-2) = 196
    • 2 x (12-2) x (16-2) = 280
    • Total = 140 + 196 + 280 = 616 blocks.
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 = 180 - angle 3
  • Angle 1 is .4(180-angle 3)
  • Angles 1 and 3 add to 90 degrees, so
  • .4(180 - angle 3) + angle 3 = 90
  • 72 - .4 angle 3 + angle 3 = 90
  • .6 angle 3 = 90 - 72 = 18
  • angle 3 = 18 / .6 = 30 degrees

Problem
Solution
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. Since these triangles are similar they are the same except for a scale factor. The ratio of the 2 sides of triangle BPQ to the similar sides of triangle ADP are:
        2/3 = 5/X
  5. X = QC = 5 x 3 / 2 = 7.5
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 12x9 = 108 sq. cm.
  2. The height of the rectangle is 4/3 the base length
  3. The larger rectangle area = 216 sq. cm.
  4. Let B = the new base length, H = new height
  5. B x 4B/3 = 216
  6. B2 = (3/4) x 216 = 162
  7. B = 12.73 cm
  8. H = (4/3) x 12.73 = 16.97 cm.
  9. H/h = 16.97/12 = 1.414
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 shown in the figure to the left.
  3. To make the reflected point C translate to point T (and the other vertices line up, too) , you must move point reflected triangle ABC down 1 unit and to the left 3 units.