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Problem of the Week
Problem B
"Try"angles

Using four straight lines, it is only possible to construct up to two non-overlapping triangles. Here are some examples:

Two lines cross to form an X. A third line joins the top two points of the X forming a triangle. A fourth line joins the bottom two points of the X forming another triangle. The two triangles are labelled 1 and 2.   Three lines form a triangle. A fourth line goes from the top vertex of the triangle to the base dividing this triangle into two smaller triangles. The two smaller triangles are labelled 1 and 2.

Using five straight lines, it is only possible to construct up to five non-overlapping triangles. Here are some examples:

Two lines cross to form an X. A third line joins the top two points of the X forming a triangle. A fourth line joins the bottom two points of the X forming another triangle. A fifth line is drawn from the base of one triangle to the base of the other, passing through the centre of the X, dividing each of the two triangles into two smaller triangles. The four triangles formed are labelled 1, 2, 3, and 4.   Five lines form a five-pointed star. The five lines are arranged so they make a pentagon in the centre with a triangle on each of the five sides of the pentagon. The five triangles are labelled 1 through 5.

Notice that the first diagram has four non-overlapping triangles and the second diagram has five non-overlapping triangles. Notice also that the diagram with five non-overlapping triangles also has a pentagon which is not counted.

  1. How many non-overlapping triangles can you make using six straight lines?

  2. How many non-overlapping triangles can you make using seven straight lines?

Trade ideas with a classmate.

Theme: Geometry & Measurement