Video Transcript
Is the line segment between the
midpoints of two sides of a triangle parallel to the other side?
To answer this question, let’s take
a triangle, which we can call triangle 𝐴𝐵𝐶. We want to consider the line
segment between the midpoints of two sides of the triangle, which is also called a
midsegment. We can draw 𝐷 and 𝐸 at the
midpoints of line segments 𝐴𝐵 and 𝐴𝐶. And as these are midpoints, we know
that there will be two pairs of congruent line segments, since 𝐴𝐷 and 𝐷𝐵 are
congruent and 𝐴𝐸 and 𝐸𝐶 are congruent. We can join the midpoints 𝐷 and 𝐸
to create the midsegment, which will be line segment 𝐷𝐸.
We now need to determine if the
midsegment 𝐷𝐸 is parallel to the third side of the triangle, which is line segment
𝐵𝐶. The important first step we take in
working out the answer is to draw a new line. We construct the line passing
through 𝐶 and a new point 𝑃 such that this new line 𝐶𝑃 is parallel to the line
segment 𝐴𝐵. We can then extend the line segment
𝐷𝐸 so that it meets the new line 𝐶𝑃 at the point 𝐹.
Now let’s consider some of the
sides and angles in the triangles 𝐴𝐸𝐷 and 𝐶𝐸𝐹. Firstly, we know that we have two
congruent sides. Since we set 𝐸 to be the midpoint
of line segment 𝐴𝐶, we can write that 𝐴𝐸 equals 𝐶𝐸. Then, because we have a pair of
parallel lines and a transversal line segment 𝐴𝐶, we can identify a pair of
alternate interior angles. As these angles are congruent, we
can write that the measures of angles 𝐷𝐴𝐸 and 𝐹𝐶𝐸 are equal. We can also observe that angles
𝐷𝐸𝐴 and 𝐹𝐸𝐶 are vertically opposite angles. So their measures are also
equal. And so because we have two pairs of
congruent angles and an included pair of sides congruent, we can write that
triangles 𝐴𝐸𝐷 and 𝐶𝐸𝐹 are congruent by the ASA criterion.
We can notice at this point that
even with these congruent triangles, we still haven’t determined if we have a pair
of parallel lines. But let’s think about what we can
determine from these congruent triangles. We can identify that the
corresponding sides 𝐴𝐷 and 𝐶𝐹 must be congruent. And of course we already created
the line segment 𝐴𝐵 such that 𝐷 is the midpoint. So 𝐵𝐷 is also congruent to
𝐴𝐷. Now we know that in the figure,
there are three congruent line segments.
The final step is to consider the
polygon 𝐵𝐶𝐹𝐷. 𝐵𝐶𝐹𝐷 is a quadrilateral, and we
have shown that the sides 𝐵𝐷 and 𝐹𝐶 are congruent. And we know that they are also
parallel because we constructed the line 𝐶𝑃 to be parallel. This property is sufficient to
prove that the shape 𝐵𝐶𝐹𝐷 is a parallelogram. And we know that parallelograms
have two pairs of opposite sides parallel. So line segments 𝐷𝐹 and 𝐵𝐶 are
also parallel.
We can therefore return to the
original question to see if the midsegment of a triangle is parallel to the third
side. And the answer is yes, because the
line segment 𝐷𝐸 is contained within the line segment 𝐷𝐹. So either line segment 𝐷𝐸 or 𝐷𝐹
can be said to be parallel to line segment 𝐵𝐶. In fact, this property that we have
just proved is one of the triangle midsegment theorems. It can be stated as the line
segment joining the midpoints of two sides of a triangle is parallel to the third
side. We can learn this property and
apply it directly in many different problems. Stating this theorem would be
sufficient for us to give the answer to the question as yes.
