Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Geometry
Problem
Hint
1) In the diagram there are two parallel line segments and a labeled line segments R that meets the line segments at a 30 degree angle. On your answer sheet, 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?

 After rotation, the 30 degree segment's angle with R is _______ degrees.
2) 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?
  1. The radius of the enlarged circle, in terms of its original radius (r) is _____ r.

  2. Because the area is increased as the square of the radius, the ratio of the new circle's area to the original one is ______

Problem
Hint
3) The isosceles trapezoid ABCD is the union of two smaller trapezoids ABFE and EFCD. AB has length 8 units, EF has length 4 units and DC has length 2 units. The area of trapezoid ABCD is 20 square units. What is the height of trapezoid EFCD? The area of a trapezoid is
(b1 + b2) h / 2 where b1 and b2 are the 2 parallel bases and h is the height.
  1. Compute the height of trapezoid ABCD using its bases and its area = ______

  2. Drop a line from point D down to line AB, making right triangle AGD. This triangle's base is ____ and its height is the same as the big trapezoid = _______.

  3. The base of EHD is ______

  4. Triangles AGD and EHD are similar so their side lengths are proportional.
  5. So we set up the ratio of the height of EHD to AGD and set it equal to the ratio of their bases: _____ for EHD and ____ for AGD:


  6. Therefore, the height of EHD (and the trapezoid EFCD) = ______
4) An isosceles triangle has vertices at
(-5, 1), (5, 1), and (0, 8). How many points lying in its interior also have only integer coordinates?

 OK, plot the triangle points and connect them, then count the integer vertices that lie inside it:


Number = _____ vertices.
5) The circumference of the bowling ball is 27 inches. If the volume of a sphere is given by
V = 4/3 r3, where r is the radius, then what is the volume of the bowling ball to the nearest tenth of a cubic inch?
Use = 3.14 in your computations or give an exact answer leaving in your answer.
  1. Compute the radius of the sphere given its circumference:_______




  2. V = ___________