Explanation
To understand this question, we will draw a right
triangle:
We draw one that is very non-isosceles so that there is no
confusion about what the smallest and second-smallest sides are. The
hypotenuse, as always is the longest side. The left side can be called the
height of the triangle and is the smallest side. The base of the triangle is
the second-smallest side. The line that we have to add bisects the smallest
side:
The "smaller triangle" that the question refers to must be
what's labeled here as ABC; it, too,
is a right triangle, since BC is
parallel to the base of the larger triangle. We want the area of this smaller triangle.
On to the data statements, separately first, as always.
Statement (1) gives the area of the larger triangle. Will
that be sufficient to get the area of the smaller triangle? It's clear that the
height of the smaller triangle is half the height of the larger triangle. There
is a more specific relationship between the two triangles, however, since they
have identical angles. They are therefore "similar triangles," meaning they
have the same shape. Since the side opposite a given angle scales proportionally
with that angle, two similar triangles, with identical angles, will have sides
of proportional lengths. The upshot is that, since the height of the smaller
triangle is half the height of the larger triangle, the base must be half of
the base of the larger triangle. Substituting
and
into the area
formula indicates that the area of the smaller triangle is
, one quarter of the larger triangle. In our
diagram, the smaller triangle doesn't quite look that small--could four of them
fit inside the larger triangle? Indeed, if we draw the situation accurately we
get:
Four triangles in one! If you aren't certain about this
conclusion, you might be able to convince yourself by analysis by cases, in
fact. Namely, if you can convince yourself that the case above of 4-in-1
triangle configuration is possible, you can conclude that it must entail
bisecting the height and the base, and then
you can convince yourself that, working the other way, when the height is
bisected by a line parallel to the base, the only possible case is the 4-in-1
case. The area of the smaller triangle is a quarter of the larger area, so it
must be 12, and we have answered the question. Sufficient.
A further side note: this is actually connected to a way
of thinking about the formula for the area of a triangle. It's no coincidence
that the formula for the area of a triangle
is half the
formula of a rectangle,
, in that any triangle can be doubled and converted
into a rectangle. For example:
(It's true also for non-right triangles. You can get out
your Tangram set and play around with two congruent triangles.)
Statement (2): Knowing the base is insufficient, because
we know nothing about the height of the smaller or bigger triangle. We can
imagine two cases, one in which the triangle has a small height and another in
which the triangle has a large height. Both cases are allowed by the data and
yield different areas, so Statement (2) is insufficient.
The correct answer is (A).
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