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FOLDING CIRCLES

FOLDING CIRCLES. Fold the edge of the circle so that one of the points on the circumference of the circle lands on the center of the circle. This fold line defines a chord of the circle. Fold a second point to the center. The resulting shape will look like and ice-cream cone.

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FOLDING CIRCLES

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  1. FOLDING CIRCLES

  2. Fold the edge of the circle so that one of the points on the circumference of the circle lands on the center of the circle. This fold line defines a chordof the circle.

  3. Fold a second point to the center. The resulting shape will look like and ice-cream cone.

  4. Fold the third point to the center. • What is the name of the shape • that has been formed? • What properties does this shape have? • Prove that this is an equilateral triangle • without using a ruler.

  5. Find the midpoint of one side of the triangle by folding one vertex of the triangle on top of a second vertex and pinching. • Then fold the third vertex • to this midpoint.

  6. What is the name of the shape that is formed? • What properties does this shape have? • Open the triangle back up. What do you notice about this new fold line, its endpoints, and its relationship to the opposite side of the triangle?

  7. Fold back to the trapezoid. How are the two bases of this trapezoid related? • If we let the area of the original triangle be 1, what is the area of the trapezoid? How do you know?

  8. Fold the vertex on the bottom right to the vertex on the top left. • What is the name of this new shape? • What properties does this shape have? • If we let the area of the original triangle be 1, what is the area of the rhombus? How do you know?

  9. Now fold the vertex on the bottom left to the vertex on the top right. • What shape has now been formed? • How does this new shape compare to the original triangle?

  10. What is the ratio of the length of the side of the large triangle to the length of the side of the small triangle? • What is the ratio of the area of the large triangle to the area of the small triangle? • If the area of the original triangle is 1, what is the area of this small triangle? How do you know?

  11. Unfold the shape back to the original triangle. Now fold the three vertices of the original triangle up to a single vertex. • What shape has been formed? • What properties does this shape have? • If the area of the original triangle is 1, what is the surface area of this pyramid?

  12. Unfold back to the original triangle. Choose a vertex of this triangle and fold it to the center of the circle. Fold the other two vertices of the triangle to the center of the circle. • What shape do you have now? Is it regular? • If the area of the original triangle is 1, what is the area of the hexagon?

  13. If you look carefully you will see a triangle inscribed within the hexagon. This triangle is surrounded by three small trapezoids. What is the area of one of these trapezoids?

  14. Fold the top halves of the triangles down across each other to make a truncated pyramid. • If the area of the original triangle is 1, what is the surface area of this truncated pyramid? How do you know? • Look at the two triangular bases. What is the linear scale factor? • What is the ratio of the area of the small triangle to the area of the large triangle?

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