Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Geometry
Problem
Solution
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?

 The sketch to the left is the rotated lines. The angle they make with R is still 30 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. Area of smaller circle = r2
  2. Area of larger circle = (1.5 r) 2
  3. Ratio of larger to smaller =
    2.25 r 2 = 2.25
    r2

Problem
Solution
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. The height of trapezoid ABCD = (8 + 2) h /2 = 20
    10 h /2 = 20; h = 4
  2. Drop a line from point D down to line AB, making right triangle AGD whose base is 3 and whose height is 4.
  3. Triangles AGD and EHD are similar so their side lengths are proportional.
  4. So we set up the ratio of the height of EHD to AGD (4) and set it equal to the ratio of their bases: 1 for EHD and 3 for AGD:
    h/4 = 1/3
  5. Therefore, the height of EHD (and the trapezoid EFCD) = 4/3
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, we have to plot this one:
If you count the integer vertices that lie within the triangle, that number is 30
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. The radius of the sphere is 27 / 2
  2. V = 4⁄3 (27/2 )3
  3. V = (27)3 / 62
  4. V = 3280.5/ 2 cu. in.