Video Transcript
In the given figure, if 𝐴𝐶 equals
four centimeters and 𝑥𝑦 equals two centimeters, which of the following is
true? Option (A) 𝐴𝐵 equals 𝐴𝐶. Option (B) 𝐵𝐴 equals 𝐵𝐶. Option (C) 𝐴𝑥 equals 𝐴𝑦. Option (D) 𝐵𝑥𝑦 is an isosceles
triangle. Or option (E) 𝑥 and 𝑦 are the
midpoints of line segment 𝐴𝐵 and line segment 𝐴𝐶.
We can begin by noting that the
markings on the figure indicate that line segments 𝑥𝑦 and 𝐴𝐶 are parallel. We are also given that the lengths
of these line segments 𝐴𝐶 and 𝑥𝑦 are four centimeters and two centimeters,
respectively. And if we realize that two is half
of four, then that means we can say that this line segment 𝑥𝑦 in the middle of
triangle 𝐴𝐵𝐶 is half the length of the base of the triangle.
That might make us think about the
triangle midsegment theorem. This theorem states that the
midsegment of a triangle is parallel to the third side and is half its length, where
the midsegment is the line joining the midpoints of two sides of a triangle. And the converse of this theorem is
also true. That is, if a line connecting two
sides of a triangle is parallel to the third side and is half its length, then it is
a midsegment. And this is what we have here in
the figure.
Line segment 𝑥𝑦 is a line
connecting two sides of a triangle. And it is parallel to the third
side and half its length. Therefore, line segment 𝑥𝑦 must
be a midsegment of the triangle 𝐴𝐵𝐶. However, none of the given answer
options have this response. So let’s see what’s equivalent to
stating that line segment 𝑥𝑦 is a midsegment.
By recalling that a midsegment is a
line joining the midpoints of two sides of a triangle, we can state that 𝑥 and 𝑦
must be the midpoints of the line segments 𝐴𝐵 and 𝐴𝐶. This is the answer that was given
in option (E).
Note that we don’t have enough
information here to prove that any of the other statements in options (A) to (D) are
true, just that 𝑥 and 𝑦 are the midpoints of the line segments 𝐴𝐵 and 𝐴𝐶.
