Hints and solutions to
Problem of the day for Wednesday, November 6
How many triangles can be drawn having any of these points as vertices?
It might be a little tough actually to draw all the triangles and
then
count them.
So we have to find a
different way to go about figuring out
how
many there are.
A triangle has three vertices, so a first crack at figuring out how many
triangles would be to count how many ways you can pick three of the nine
vertices in the figure.
For students who know about such things, this number
is C(9,3) = 84. Or, you can reason that there are 9 ways to pick the first
vertex, then 8 ways to pick the second, and 7 to pick the third. So that's
9 x 8 x 7 ways to pick the three vertices in order. But the order
doesn't matter, so you have to divide by the six orders in which you could
have chosen the same three vertices. That makes 84.
But not all 84 choices of three points are the vertices of a triangle.
There
are a few ways to choose three collinear points (in other words,
three points that lie on one straight line). But these are easy to count.
There are three vertical lines with three of the points, three horizontal
lines,
and two diagonal lines.
Thus, 8 of the 84 choices
of three points do not
give us a triangle.
How many triangles can be drawn having any of these points as vertices?
