### Video Transcript

In the given figure, which of the
following is true? Option (A) πΈ is the midpoint of
line segment πΉπΊ. Option (B) πΉ is the midpoint of
line segment π΄π·. Option (C) πΉπΊ equals one-half
π΄π΅. Or option (D) πΆπ· equals one-half
π΄π΅.

Letβs begin by having a look at the
figure. We could observe that there are two
pairs of congruent line segments, with the first pair being the line segments π΄πΈ
and πΈπΆ. The second pair of congruent line
segments are π΅πΊ and πΊπΆ. Therefore, we can note that points
πΈ and πΊ must be the midpoints of the line segments π΄πΆ and π΅πΆ,
respectively. And because we have these
midpoints, we can apply the triangle midsegment theorem.

The line segment connecting the
midpoints of two sides of a triangle is parallel to the third side and is half its
length. So, if we consider triangle π΄π΅πΆ,
the line segment πΈπΊ connecting the midpoints of the two sides must be parallel to
the third side, π΄π΅. And we were already given that line
segments π΄π΅ and πΆπ· were parallel, so we have three parallel line segments.

Now, letβs consider which of the
four statements are true, taking option (A) first. πΈ is the midpoint of line segment
πΉπΊ. Now, we have established that πΈ is
a midpoint. But itβs the midpoint of line
segment π΄πΆ. Line segment πΉπΊ is this line
segment in the middle of the figure. And we donβt have any information
to prove that πΈ is the midpoint of line segment πΉπΊ. Therefore, we canβt say that option
(A) is true.

Next, we can consider the statement
in option (B). πΉ is the midpoint of line segment
π΄π·. We can find point πΉ and line
segment π΄π· on the left side of the diagram. If we consider this line segment as
a side of triangle π΄πΆπ·, we can determine something about this line segment. Recalling the theorem that the line
segment passing through the midpoint of one side of a triangle that is also parallel
to another side of the triangle bisects the third side of the triangle, we can
observe that the line segment πΈπΉ is a line segment passing through the midpoint of
one side of the triangle and it is parallel to another side. Therefore, the third side, which is
the line segment π΄π·, is bisected by line segment πΈπΉ. That means that point πΉ is the
midpoint of line segment π΄π·. And so, the statement in option (B)
is true.

However, itβs worth checking if
either of options (C) or (D) is also a true statement. Option (C) says that πΉπΊ equals
one-half π΄π΅. To see if this is true or not, we
can look at triangle π΄π΅πΆ which is colored in green. By the first theorem here, since we
have the midsegment πΈπΊ in this triangle, we know that πΈπΊ must be half the length
of the parallel side π΄π΅. But the given statement doesnβt say
that πΈπΊ is one-half π΄π΅; it says that πΉπΊ is. And we can see that the points πΉ
and πΈ donβt lie on the same position. So, statement (C) is not true.

Finally, we can take the last
statement that πΆπ· equals one-half π΄π΅. Line segment πΆπ· is here at the
bottom of the figure. But we canβt apply any of the
triangle midsegment theorems or any other theorems here to tell us that this is a
true statement.

Therefore, the only statement that
we can say is true is that in option (B): πΉ is the midpoint of line segment
π΄π·.