Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2016 Grade 7 Geometry
Problem
Solution
1) Find the perimeter of the figure to the nearest centimeter. Each circular section has a radius of 13 cm.
  1. Note that the 2 quarter-circles on the bottom of the figure, when turned around, complete each of the 2 circles, therefore the perimeter (circumference) of the 2 circles is:
        C = d
        C = 26
        2C = 52 cm.
  2. The additions are the 2 radii of the 2 quarter circles =
    2 x 13 = 26 cm.
  3. Total = 52 x 3.14 + 26 = 189.28 = 189 cm. (rounded)
2) The ratio of the angles of a triangle is 4:7:9. What is the measure of largest angle?
  1. Add the parts together: 4 + 7 + 9 = 20 parts
  2. The angles are all multiples of
    180/20 = 9 degrees.
  3. The largest angle is 9x9 = 81 degrees
3) In rectangle ABCD, X is the midpoint of segment AB and Y is the midpoint of segment CD. The area of the shaded portion is 5.75 square inches. What is the area of the rectangle ABCD?
 The lines XC and BY cut the rectangle XBCY into 4 equal parts.
Since XBCY is half the rectangle, the area of ABCD is
8 x 5.75 = 46 sq. in.

Problem
Solution
4) A 110o section is taken from a circle with a radius of 4 inches and folded to create a cone by connecting the straight edges together. What is the radius of this cone? (Use 3.14 for pi)
(Round your answer to the nearest tenth of an inch)
 See the figure to the left.
  1. The circumference of the circle is C = 8 inches.
  2. The portion of the circumference that makes the cone is:
        110/360 x 8 = 22⁄9 in.
  3. This is now the circumference of the base of the cone, so the radius of this cone is:
        r = C/(2) =
        r = 22⁄9 / (2)
        r = 11⁄9 = 1.2 inches (rounded)
5) In triangle XYZ, segment VW is parallel to XY. What is the measure of angle XVW?
  1. The measure of angle YXZ is 180 - 90 - 52 = 38 degrees.
  2. Triangles XYZ and VWZ are similar triangles because their hypotenuses are parallel, so angle WVZ is equal to angle YXZ = 38 degrees.
  3. Angle XVW and angle WVZ add to 180 (a straight line) so angle XVW =
    180 - 38 = 142 degrees