Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 8 Geometry
Problem
Hint
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 ____ degrees every hour while the minute hand moves ____ degrees in that same amount of time.
  2. So, the difference in their rates is _____ deg/hour, meaning the angle between the hands increases at the rate of _____ deg/hour.

  3. So, starting at 30 degrees separation and moving H hours, the separation S would be:
    S = ____________________

  4. For a separation of 85 degrees, this would take:
    H = ______ minutes, so the time would be _____________
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? Plot your points on the blank graph to the left.
  1. Call the given points A and B.
  2. Construct line segment AC which is half the length of the base AB. Plot it on the blank graph.
  3. Translate AC down 1 space in y to put point A at the origin. Plot it. Call this new C point C2.
  4. Rotate point C2 both clockwise (↷) and counterclockwise (↶) 90 degrees, making 2 new points C3 and C4 =

    C3 = (___,___), C4 = (___,___)

  5. Now translate these 2 segments by (+3,+2) to put point A where C started. This makes the two segments the heights of the isosceles triangles. Plot these 2 points.
The rotated and translated point C (both clockwise and counterclockwise) will be the possible vertices of the isosceles triangle = (___,___) and (___,___)

Problem
Hint
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
        _______ degrees.

  3. 3 of them = ______ degrees.

  4. So angle ABC, which is a full circle - 3 pentagon interior angles = ___________ degrees.

Problem
Hint
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 90 degrees around the origin counterclockwise (↶), each coordinate (x,y) becomes (-y,x):
    Original ABCD =

    Rotated ABCD =

  2. Translate each of these points up 1 (in y) and 1 to the right (in x):
    TUVW =

  3. Plot these 8 points on the blank graph to the right
  4. The overlap is _________________
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____ degree angle because the line cutting the 2 parallel lines defines both those angles
  2. The other angle of that triangle is
    ______ degrees.

  3. That angle is supplementary with angle x is Therefore, angle x = _______ degrees.