### Video Transcript

If the areas of triangle πΏππ΄ and
triangle ππ΄πΊ are the same, which of the following must be true? Is it (A) ππΏ equals ππΊ? (B) Line segment ππΏ is parallel
to line segment ππΊ. Is it (C) π΄π equals π΄πΊ? (D) ππΊ equals ππΏ. Or (E) line segment ππΊ is
parallel to line segment ππΏ.

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 itβs certainly not. Weβre told that the areas of
triangle πΏππ΄ and triangle ππ΄πΊ are the same. Letβs highlight these in pink and
yellow, respectively. This is triangle πΏππ΄, and we
have triangle ππ΄πΊ here.

And so letβs 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 weβre going to do to be
able to use one of these definitions is split our triangles up.

If we split triangle πΏππ΄ and
ππ΄πΊ as shown by adding the line segments π·π and πΊπΈ, respectively, then we can
begin by considering a shared vertex π΄. This corresponds to the second
definition. These triangles share the same
vertex. And so weβll be able to say that
the area of triangle π΄πΊπΈ will be equal to the area of triangle π΄π·π if their
bases are equal length and they lie along the same line. Well, in fact, the dashes on line
segment πΊπΈ and π·π indicate that theyβre 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 π΄πΊπΈ is equal to the area of triangle π΄π·π.

Well, since the areas of our larger
triangles are the same, this in turn means that the area of triangle πΊπΈπ must be
equal to the area of triangle π·ππΏ. These triangles donβt have a shared
vertex. But they do have bases of equal
length, which lie on the same line. Thatβs πΊπΈ and π·π. And so we can use our first
definition.

Now, it doesnβt really matter that
πΊπΈ and π·π are different line segments. The simple fact that they lie on
the same line and theyβre equal in length is good enough as saying that they share
the same base. And so we can say that for triangle
πΊπΈπ to have an equal area to triangle π·ππΏ, the vertices π and πΏ β thatβs
these two β must lie on a line parallel to the base. And so the lines passing through
πΊπΈπ·π and ππΏ must be parallel. If we go to our options, we see
that corresponds to option (D). The line segment ππΏ must be
parallel to the line segment ππΊ. The answer is (B).