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Problem

The figures in the diagram below are similar. They each contain a circle and some straight lines.

  1. Suppose new figures are created by removing all straight lines which lie outside the circle from 75\(\%\) of the figures.

    1. Draw a diagram showing what one of the new figures would look like.

    2. How many new figures are created?

    3. How many of the original figures remain unchanged?

  2. Now suppose that the circle is removed from \(\frac{2}{3}\) of the new figures created in part a).

    1. How many of the new figures from part a) will still contain a circle?

    2. Draw a diagram of one of the new figures with the circle removed. Then name all the geometric shapes formed by the straight lines in this figure.

There are 16 figures. Each figure is formed as follows: A hexagon is drawn with a circle passing through all of its six vertices. From each vertex of the hexagon, two straight lines are drawn outside the circle. From one vertex of the hexagon, two straight lines are drawn inside the circle joining it to two other vertices that are side by side.

Solution

    1. A typical new figure would look like this:

      A hexagon with a circle passing through its vertices. From one vertex, two straight lines are drawn inside the hexagon joining it to two other vertices that are side by side.

    2. There are \(16\) figures in total. Since \(75\%=\frac{75}{100}=\frac{3}{4}=\frac{12}{16}\), that tells us \(12\) new figures were created.

    3. There are \(16\) figures in total and \(12\) of them were changed. So \(16-12=4\) of the original figures remain unchanged.

    1. There were \(12\) new figures created in part a). Since \(\frac{2}{3}=\frac{8}{12}\), that tells us 8 of the figures have no circle, so \(12-8=4\) will still have a circle.

    2. A typical new figure would look like this:

      A hexagon. From one vertex, two straight lines are drawn inside the hexagon joining it to two other vertices that are side by side.

      It contains a pentagon, a hexagon, two trapezoids, and two triangles, as shown below.

A pentagon formed by four sides of the hexagon and one of the lines inside the hexagon.The entire hexagon.Two identical trapezoids each formed by three sides of the hexagon and one of the lines inside the hexagon.Two triangles. One triangle is formed using one side of the hexagon and both lines inside the hexagon. The other triangle is formed using two sides of the hexagon and one of the lines inside the hexagon.