Video Transcript
Fill in the blank. If the area of a triangle whose
vertices are ℎ, zero; six, zero; and zero, three is nine square units, then ℎ
equals what. Is it option (A) zero or
negative 12? Option (B) zero or 12. Option (C) negative six or
six. Or is it option (D) 12 or
negative 12?
In this question, we’re given
the coordinates of the vertices of a triangle which contains a single unknown
coordinate and the area of a triangle. We want to use this to find the
possible values of the unknown. We could do this by using any
method for finding the area of a triangle. However, since we are given the
vertices, we will do this by using determinants.
We recall that the area of a
triangle can be calculated by using the formula one-half times the absolute
value of the determinant of the three-by-three matrix 𝑥 sub one, 𝑦 sub one,
one, 𝑥 sub two, 𝑦 sub two, one, 𝑥 sub three, 𝑦 sub three, one, where 𝑥 sub
𝑖 and 𝑦 sub 𝑖 are the coordinates of each vertex of the triangle for each
value of 𝑖.
We can substitute the
coordinates of the triangle and the area being equal to nine into this formula
to obtain the following equation. We can solve for ℎ by expanding
and isolating ℎ on one side of the equation. We can evaluate the determinant
in many ways. The easiest is to note that the
second column contains two zeros. So expanding over this column
will only have a single nonzero term.
We obtain one-half the absolute
value of negative three times ℎ minus six. We can simplify this further by
taking the factor of negative three out of the absolute value. This becomes a factor of
positive three to give the following. This is equal to the area of
the triangle, which is nine. We can solve for ℎ by
rearranging. We divide both sides of the
equation by three over two. This yields that six is equal
to the absolute value of ℎ minus six.
We can then solve this absolute
value equation by constructing two equations. Either six equals ℎ minus six
or six equals negative one times ℎ minus six. Solving each equation gives us
that ℎ is equal to 12 or ℎ equals zero. We can then see that this is
given by option (B).
