Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 7 Geometry
Problem
Solution
1) A circle is enlarged by increasing its radius by 50%. What is the ratio of the area of the enlarged circle to the area of the original circle? You may express your answer as a fraction.
(Hint: you can work with any size radius.) 		Use 1 as the radius of the original circle
  1. Area of the original circle =
  2. Area of the enlarged circle =
    (3⁄2) 2 = 9⁄4
  3. The ratio of the larger to the smaller is:
    9⁄4 = 9⁄4
       
2) In the diagram there are two parallel line segments and a labeled line segment R that meets the line segments at a 30 degree angle. Sketch what the result is when you reflect the two parallel line segments across R. What are the measures of the new angles formed between the reflected line segments and line segment R? The sketch to the left is the rotated lines. The angle they make with R is still 30 degrees
3) Which of the following nets will correctly fold to make a triangular prism with one pointed tetrahedron side?
(Think of a pointed three sided stake.)
1. Visualize folding these flat shapes into the triangular prism with the tetrahedron end.
2. Some of them will have shapes that overlap.
3. The ones that fold correctly are b & c

Problem
Solution
4) In triangle ABC shown, AB is perpendicular to CB, and DE is parallel to AC. The angle at vertex A measures 42 degrees. What is the measure of angle DEC with the label y?
  1. If angle A is 42 degrees, the angle at C is 90 - 42 = 48 degrees.
  2. If DE and AC are parallel, then triangles ABC and DBE are similar, meaning their angles are the same but their sizes are different.
  3. Therefore angle DEB is the same as angle C = 48 degrees.
  4. DEC and DEB add to 180 degrees, so angle DEC = 180 - 48 = 132 degrees
5) When you translate triangle ABC 5 units east, 4 units north, 7 units west and finally 9 units south, what are the final coordinates for vertex B? The translations are:
east: x+5, north:y+4, west: x-7 and south: y-9 =
x+5-7 and y+4-5, so the point at B(3,3) becomes (3+5-7,3+4-5) = (1,-2)

See figure to the left.