Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 8 Geometry
Problem
Solution
1) Jazmin is playing around with possible side lengths for isosceles triangles. She constructs one that has one side twice the length of another side. The lengths are whole numbers. What is the ratio of the area of the smallest possible isosceles triangle that fits this description to the area of the second smallest isosceles triangle that fits this description? We must use the pythagorean theorem to determine the heights of the triangles.
  1. The short side of the isosceles triangle cannot be the duplicated side because that would make the sides 1,1 and 2, which do not make a triangle.
  2. Therefore, the smallest triangle that can be made with whole number sides is 2,2,1
  3. The next smallest triangle's sides are 4,4,2
  4. The height of the smallest triangle is
    H = √ 2 2 - 0.5 2)  = √3.75 = 1.936 ft
  5. The area of that triangle =
    1.936 x 1 x .5 = .968 sq. ft.
  6. The height of the larger triangle is
    H = √ 4 2 - 1 2)  = √15 = 3.873 ft
  7. The area of that triangle =
    3.873 x 2 x .5 = 3.873 sq. ft.
  8. The ratio of their 2 areas is
    0.968/3.873 = 0.25 = 1/4
2) Rhonda wants to build a pendulum clock with the pendulum enclosed in a box with a glass door. Her pendulum will swing through 30 degrees and be 3 feet long. She's decided to use the length of the arc the pendulum swings through as the width for the box. What will the width be to the nearest tenth of a foot? (The arc is a section of the circumference determined by the angle.)
= 3.14
  1. The circumference of a circle with a 3 foot radius =
    6
  2. 30 degrees is 30/360 = 1/12 of the circumference = 0.5
  3. The arc length = 3.14/2 = 1.57 = 1.6 feet

Problem
Solution
3) Tori measures the edges of a toy pyramid with square base and finds they are all 6 inches long. She wants to find the area of one of the triangular faces. What is the height, h, of one of the triangles? Give an exact answer or round to the nearest tenth of an inch. Figure is not to scale. We must use the pythagorean theorem to determine the heights of the triangles.

    H = √ 6 2 - 3 2  = √27
4) Reflect triangle ABC over the line l and then reflect the result over the y-axis. Where do A, B, and C go under the two reflections? Give the coordinates for each one.
  1. The original triangle points are:
    A = (2,2), B = (7,5) and C = (4,1)
  2. The reflected image of a point (x,y) across the line y = x is achieved by switching the x and y values:
    Reflection of (x,y) around y=x = (y,x), so:
    Reflected A,B,C = (2,2), (5,7) and (1,4)
  3. A reflection of a point (x,y) over the y-axis is (-x,y), so
    Reflected A,B,C around the y-axis =
    A = (-2,2), B = (-5,7) and C = (-1,4)
5) In the picture is a representation of stretchable grabbing tongs. Segments AD, CF and EH are parallel to each other. Segments BC, DE and FG are parallel to each other. Angle AMB is 30 degrees. What is angle NFP? NFP is the supplement of 30 degrees = 150 degrees