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Problem of the Week
Problem A and Solution
Tracking Triangles

Problem

A regular polygon is a closed shape where all the side lengths are the same. Pauline draws lines inside regular polygons according to the following rules.

  1. The lines must connect two vertices that are not beside each other.

  2. The lines must be straight and cannot cross.

Pauline continues to draw lines until she cannot draw any more. At this point, the inside of her polygon will be made up entirely of triangles. For example, after drawing lines in a square she creates \(2\) triangles, and after drawing lines in a regular pentagon she creates \(3\) triangles, as shown.

A square with a dashed line along one of its diagonals.     A pentagon with two dashed lines inside. The lines start at one vertex and end at two different vertices on the opposite side of the pentagon.

  1. Notice that if Pauline had drawn lines between different pairs of vertices in the square and the regular pentagon, the resulting diagrams would have been rotations or reflections of the diagrams above, but would otherwise have been the same. Is it possible for Pauline to draw lines in a regular hexagon and create more than one diagram, which cannot be obtained from the others by a rotation or reflection? Use the hexagons below to test it out.

  2. Look at the number of triangles Pauline creates in a square, pentagon, and hexagon. Use this to predict the number of triangles Pauline creates in an octagon, and then check to see if you are correct.

Not printing this page? Try our interactive worksheet.

Solution

  1. There are three different diagrams that can be created by drawing lines in a regular hexagon. They are shown below.

    A hexagon with three dashed lines inside. The lines start at one vertex and end at three different vertices on the opposite side of the hexagon. A hexagon with three dashed lines inside. One line starts at the left vertex of the top (horizontal) side and ends at the left vertex of the bottom (horizontal) side. Another line starts at the left vertex of the bottom side and ends at the right vertex of the top side. Another line starts at the right vertex of the top side and ends at the right vertex of the bottom side. A hexagon with three dashed lines inside. The lines connect three vertices of the hexagon (every other vertex when moving around the shape) making an equilateral triangle.

    All other possible diagrams are reflections or rotations of one of these three.

  2. Pauline creates \(2\) triangles in a square, \(3\) triangles in a pentagon, and \(4\) triangles in a hexagon. It appears as though the number of triangles is always \(2\) less than the number of sides in the polygon. Then we can predict that Pauline can create \(8-2=6\) triangles in an octagon. In fact this is true. Two examples are shown.

    An octagon with five dashed lines inside. The lines all start at one vertex and end at the five different vertices that are not directly beside the starting vertex. An octagon has eight sides. Two of these sides are horizontal, with one at the top of the shape and one at the bottom, and two of these sides are vertical, with one at the left side of the shape and one at the right side. Five connected dashed lines are drawn between vertices forming a zig zag path inside the octagon. The path moves through these vertices, in order: top vertex on left side, left vertex on bottom side, left vertex on top side, right vertex on bottom side, right vertex on top side, bottom vertex on right side.

Teacher’s Notes

It takes some advanced mathematics to actually prove that in any regular polygon the maximum number of triangles we can draw according to the rules is always \(2\) less than the number of sides of the polygon. However, we can informally convince ourselves that it is possible to draw at least this many triangles. In general, one way we can draw the triangles inside the polygon is to pick one vertex as the end point of all the lines and draw a line to each of the other vertices in the polygon that are not adjacent to that vertex. If we examine that pattern, we see we always end up with the number of triangles being \(2\) less than the number of sides in the polygon.

Remembering this pattern can help us determine another feature of a polygon. It is known that the sum of the interior angles a triangle is always \(180^\circ\). You can test this by drawing many different triangles and measuring the interior angles within them. This can also be proven true for all triangles without having to measure specific angles, but again we need some more rules of geometry to do so. However, knowing this we can figure out the sum of the interior angles of any polygon by using the pattern from this problem. For example, since we can draw \(4\) triangles inside a hexagon, then the sum of the interior angles of a hexagon must be \(4\times 180^{\circ} = 720^\circ\). Similarly, the sum of the interior angles of a icosagon (a 20-sided polygon) is \(18 \times 180^{\circ}= 3240^\circ\). In general, the sum of the interior angles of a polygon with \(n\) sides is \((n - 2)\times 180^{\circ}\).