Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 8 Geometry
Problem
Solution
1) Jodi looks at an analogue clock when it is exactly 11:00 am. The angle between the hands is 30 degrees. What time is it when the angle between the hands first hits 85 degrees? Note: both hands move at different constant rates.
  1. The hour hand moves 30 degrees every hour while the minute hand moves 360 degrees in that same amount of time.
  2. So, the difference in their rates is 360 deg/hour - 30 deg/hour = 330 deg/hour, meaning the angle between the hands increases at the rate of 330 deg/hour.
  3. So, starting at 30 degrees separation and moving H hours, the separation S would be:
    S = 30 deg + (330 deg/hour) x H hours
  4. For a separation of 85 degrees, this would take:
    85 = 30 deg + 330 H degrees
    h = 55/330 = 0.166667 hour = 10 minutes, so the time would be 11:10
2) The base of an isosceles triangle has vertices at (0,1) and (6,3) as shown. Its height is half its base. What can the coordinates of its vertex be?

Figure 1
Figure 2

Figure 3
Figure 4
  1. Construct line segment AC which is half the length of the base AB. See figure 1
  2. Translate AC down 1 space to put point A at the origin. Call this line segment A2 C2 = (0,0) and (3,1). (See figure 2)
  3. Rotate point C2 both clockwise (↷) and counterclockwise (↶) 90 degrees
    • Counterclockwise rotation of C2: (x,y) becomes (-y,x), so (3,1) becomes (-1,3).
    • Clockwise rotation of C: (x,y) becomes (y,-x), so (3,1) becomes (1,-3).
    See figure 3 for these 2 new line segments.
  4. Translate these 2 segments by (+3,+2) to put point A2 where C started. This makes the two segments the heights of the isosceles triangle. See figure 4.
The rotated and translated point C (both clockwise and counterclockwise) will be the possible vertices of the isosceles triangle = (2,5) and (4,-1)

Problem
Solution
3) Avery wants to create a net that he can cut out and fold to make a regular dodecahedron. A dodecahedron is made up of 12 pentagons. Three of them come together at a vertex. What is the left over angle, angle ABC, that will create the point at vertex B once it is folded? See diagram.
  1. The interior angles of a regular polygon are: 180 (n - 2) / n
    where n is the number of sides.
  2. For our pentagon, the interior angles are each
        180x3 ÷ 5 = 108 degrees.
  3. 3 of them = 324 degrees.
  4. So angle ABC, which is a full circle - 3 pentagon interior angles = 360 - 324 =
        36 degrees
4) Rectangle PQRS is the image of rectangle ABCD after a rotation of 90 degrees counterclockwise (↶) about the origin followed by a translation 1 unit up and 1 unit to the right. Rectangle TUVW is the image of rectangle ABCD after a translation 1 unit up and 1 unit to the right followed by a rotation of 90 degrees counterclockwise about the origin. Clearly plot the set of points that is the intersection of rectangle PQRS with rectangle TUVW.

Figure 1
  1. To rotate points around the origin counterclockwise, each coordinate (x,y) becomes (-y,x):
    Original ABCD = (0,0), (0,2), (-1,2), (-1,0)
    Rotated ABCD =  (0,0), (-2,0), (-2,-1), (0,-1)
  2. Translate each of these points up 1 and 1 to the right:
    PQRS = (1,1), (-1,1), (-1,0), (1,0)
  3. Rectangle TUVW:
    • ABCD = (0,0), (0,2), (-1,2), (-1,0)
    • Translate 1 up and 1 to the right:
      TUVW = (1,1), (1,3), (0,3), (0,1)
    • Rotate 90 degrees counterclockwise:
      (x,y) becomes (-y,x):
      TUVW = (-1,1), (-3,1), (-3,0), (-1,0)
  4. These 2 rectangles are plotted in figure 2 to the left
The only overlap is the line from (-1,1) to (-1,0)

Problem
Solution
5) In the drawing below, the two lines that look parallel are parallel. With the given information can you determine the measure of the angle labeled x? If so what is it? If not, which of the labeled angles, 1, 2, 3, or 4, do you need to know to determine x?
  1. Angle 1 is the same measure as the 55 degree angle because the line cutting the 2 parallel lines defines both those angles
  2. The other angle of that triangle is
    180 - 74 - 55 = 51 degrees
  3. That angle is supplementary with angle x is Therefore, angle x = 180 - 51 = 129 degrees