Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 7 Geometry
Problem
Solution
1) Trapezoid ABCD is similar to trapezoid EBFG. Find AD. Similar figures have the same shape and differ in dimensions by a scale factor.
  1. The scale factor is AB / EB = 14/8 = 7/4
  2. Side EG corresponds to side AD in the larger trapezoid.
  3. AD = EG x 7 / 4 = 4 x 7 / 4 = 7
2) In the figure, line P is parallel to line R. Angles of measures 40 degrees and 75 degrees are marked. What is the measure of angle x?
  1. When a line cuts parallel lines, the acute angles made with the parallel lines are the same, so angle A is equal to 40 degrees.
  2. A and x add to 180 degrees, so angle x = 140 degrees

Problem
Solution
3) The points of an isosceles triangle are A=(2,3), B=(3,1), and C=(1,1). If you rotate the triangle 180 degrees about the origin and then reflect the triangle across the x-axis, what are the new coordinates for A,B, and C?
 The figure to the right shows the points after 180 degrees rotation about the origin.


The bottom figure shows the points after a subsequent rotation about the x-axis.
Therefore the new coordinates are
A(-2,3), B(-3,1), and C(-1,1)
4) If a square cake with side length 8 inches serves 4 people, how many people would be able to get an equivalent serving from a 12 inch diameter circular cake? Both cakes are the same height.
  1. The area served by 1 person is 8x8/4 = 16 sq. inches
  2. The area of a 12 inch diameter circular cake =
        area = 62 = 36
  3. The number of people this cake would serve is:
        (36 ) / 16 = 9⁄4 =
    7 people
5) A circular cone can be created by joining the two straight edges of the shown section of a circle with central angle 90 degrees and radius 5 centimeters. What is the diameter of the created cone? (The diameter of a cone is the diameter of the circular opening where it is the widest.)
  1. The length of the quarter-circle arc =
    10 /4 = 5⁄2 cm.
  2. This length is the circumference of the base of the cone, so the diameter of the cone is:
    D = C / =
    5⁄2 / = 2.5 cm.