If the areas of triangle 𝐿𝑁𝐴 and
triangle 𝑌𝐴𝐺 are the same, which of the following must be true? Is it (A) 𝑌𝐿 equals 𝑁𝐺? (B) Line segment 𝑌𝐿 is parallel
to line segment 𝑁𝐺. Is it (C) 𝐴𝑁 equals 𝐴𝐺? (D) 𝑌𝐺 equals 𝑁𝐿. Or (E) line segment 𝑌𝐺 is
parallel to line segment 𝑁𝐿.
Now, be a little bit careful. Each of these lines is drawn on a
two-dimensional plane. This is not a 3D shape. It does look a little bit like a
square-based pyramid, but it’s certainly not. We’re told that the areas of
triangle 𝐿𝑁𝐴 and triangle 𝑌𝐴𝐺 are the same. Let’s highlight these in pink and
yellow, respectively. This is triangle 𝐿𝑁𝐴, and we
have triangle 𝑌𝐴𝐺 here.
And so let’s recall what we know
about the equality of the areas of two triangles. We know that the areas of two
triangles will be equal if these triangles share the same base and their vertices
lie on a straight line parallel to the base, or if the triangles share the same
vertex and have bases of equal length along the same line. And so what we’re going to do to be
able to use one of these definitions is split our triangles up.
If we split triangle 𝐿𝑁𝐴 and
𝑌𝐴𝐺 as shown by adding the line segments 𝐷𝑁 and 𝐺𝐸, respectively, then we can
begin by considering a shared vertex 𝐴. This corresponds to the second
definition. These triangles share the same
vertex. And so we’ll be able to say that
the area of triangle 𝐴𝐺𝐸 will be equal to the area of triangle 𝐴𝐷𝑁 if their
bases are equal length and they lie along the same line. Well, in fact, the dashes on line
segment 𝐺𝐸 and 𝐷𝑁 indicate that they’re the same length. And we can quite clearly see from
the picture that they do lie along the same line. This must mean that the area of
triangle 𝐴𝐺𝐸 is equal to the area of triangle 𝐴𝐷𝑁.
Well, since the areas of our larger
triangles are the same, this in turn means that the area of triangle 𝐺𝐸𝑌 must be
equal to the area of triangle 𝐷𝑁𝐿. These triangles don’t have a shared
vertex. But they do have bases of equal
length, which lie on the same line. That’s 𝐺𝐸 and 𝐷𝑁. And so we can use our first
Now, it doesn’t really matter that
𝐺𝐸 and 𝐷𝑁 are different line segments. The simple fact that they lie on
the same line and they’re equal in length is good enough as saying that they share
the same base. And so we can say that for triangle
𝐺𝐸𝑌 to have an equal area to triangle 𝐷𝑁𝐿, the vertices 𝑌 and 𝐿 — that’s
these two — must lie on a line parallel to the base. And so the lines passing through
𝐺𝐸𝐷𝑁 and 𝑌𝐿 must be parallel. If we go to our options, we see
that corresponds to option (D). The line segment 𝑌𝐿 must be
parallel to the line segment 𝑁𝐺. The answer is (B).