Video Transcript
In the given figure, which of the
following is true? Option (A) 𝐸 is the midpoint of
line segment 𝐹𝐺. Option (B) 𝐹 is the midpoint of
line segment 𝐴𝐷. Option (C) 𝐹𝐺 equals one-half
𝐴𝐵. Or option (D) 𝐶𝐷 equals one-half
𝐴𝐵.
Let’s 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 𝐴𝐸
and 𝐸𝐶. The second pair of congruent line
segments are 𝐵𝐺 and 𝐺𝐶. Therefore, we can note that points
𝐸 and 𝐺 must be the midpoints of the line segments 𝐴𝐶 and 𝐵𝐶,
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 𝐴𝐵𝐶,
the line segment 𝐸𝐺 connecting the midpoints of the two sides must be parallel to
the third side, 𝐴𝐵. And we were already given that line
segments 𝐴𝐵 and 𝐶𝐷 were parallel, so we have three parallel line segments.
Now, let’s consider which of the
four statements are true, taking option (A) first. 𝐸 is the midpoint of line segment
𝐹𝐺. Now, we have established that 𝐸 is
a midpoint. But it’s the midpoint of line
segment 𝐴𝐶. Line segment 𝐹𝐺 is this line
segment in the middle of the figure. And we don’t have any information
to prove that 𝐸 is the midpoint of line segment 𝐹𝐺. Therefore, we can’t say that option
(A) is true.
Next, we can consider the statement
in option (B). 𝐹 is the midpoint of line segment
𝐴𝐷. We can find point 𝐹 and line
segment 𝐴𝐷 on the left side of the diagram. If we consider this line segment as
a side of triangle 𝐴𝐶𝐷, 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 𝐸𝐹 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 𝐴𝐷, is bisected by line segment 𝐸𝐹. That means that point 𝐹 is the
midpoint of line segment 𝐴𝐷. And so, the statement in option (B)
is true.
However, it’s worth checking if
either of options (C) or (D) is also a true statement. Option (C) says that 𝐹𝐺 equals
one-half 𝐴𝐵. To see if this is true or not, we
can look at triangle 𝐴𝐵𝐶 which is colored in green. By the first theorem here, since we
have the midsegment 𝐸𝐺 in this triangle, we know that 𝐸𝐺 must be half the length
of the parallel side 𝐴𝐵. But the given statement doesn’t say
that 𝐸𝐺 is one-half 𝐴𝐵; it says that 𝐹𝐺 is. And we can see that the points 𝐹
and 𝐸 don’t lie on the same position. So, statement (C) is not true.
Finally, we can take the last
statement that 𝐶𝐷 equals one-half 𝐴𝐵. Line segment 𝐶𝐷 is here at the
bottom of the figure. But we can’t 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): 𝐹 is the midpoint of line segment
𝐴𝐷.
