Video Transcript
Given that line segment 𝐴𝑀 is a
median of triangle 𝐴𝐵𝐶 and that 𝐵𝑀 is less than 𝐴𝑀, what type of angle is
angle 𝐵𝐴𝐶? Option (A) acute, option (B)
obtuse, option (C) right. Or is it option (D) reflex?
In this question, we are given some
information about a triangle 𝐴𝐵𝐶. And we want to use this information
to determine the type of angle 𝐵𝐴𝐶. To do this, let’s start by adding
the given information onto the diagram.
First, we are told that line
segment 𝐴𝑀 is a median of triangle 𝐴𝐵𝐶, which means it bisects the side of the
triangle opposite vertex 𝐴. This is already shown on the
figure, since 𝐵𝑀 is equal to 𝐶𝑀. Next, we are told that 𝐵𝑀 is
shorter than 𝐴𝑀. We cannot directly represent this
on the figure. However, we can label 𝐵𝑀 and 𝐴𝑀
as shown. We want to determine the type of
angle 𝐵𝐴𝐶. We can add this angle onto the
diagram to see that it is the internal angle of triangle 𝐴𝐵𝐶 at vertex 𝐴.
It is worth noting that this is
enough to eliminate option (D), since internal angles in a triangle must have
measure less than 180 degrees. So the angle cannot be a reflex
angle.
Since we are given the comparison
of two side lengths in the figure, we can apply the angle comparison theorem in
triangles to determine information about the measures of the angles.
We recall that this tells us in a
triangle 𝑋𝑌𝑍 if side 𝑋𝑌 is longer than side 𝑌𝑍, then the angle opposite 𝑋𝑌
has larger measure than the angle opposite 𝑌𝑍. So the measure of angle 𝑍 is
greater than the measure of angle 𝑋. We can see that in triangle 𝐴𝑀𝐵,
the side 𝐴𝑀 is longer than the side 𝐵𝑀. So the angle opposite 𝐴𝑀 has
larger measure than the side opposite 𝐵𝑀. Hence, the measure of angle 𝐵 is
greater than the measure of angle 𝐵𝐴𝑀.
We can also apply this theorem to
triangle 𝐴𝑀𝐶 by noting that 𝐴𝑀 is a median of the triangle. So 𝑀𝐶 has length 𝐵𝑀. This means that 𝐴𝑀 is longer than
𝐶𝑀. So the angle opposite 𝐴𝑀 has
larger measure than the angle opposite 𝐶𝑀. Hence, the measure of angle 𝐶 is
greater than the measure of angle 𝐶𝐴𝑀.
This allows us to construct an
inequality involving the measure of angle 𝐵𝐴𝐶 by noting that the two angles on
the right-hand side of these inequalities combine to make angle 𝐵𝐴𝐶. Therefore, if we add the left- and
right-hand sides of the two inequalities together, then we get that the measure of
angle 𝐵 plus the measure of angle 𝐶 is greater than the measure of angle
𝐵𝐴𝐶.
This inequality now includes all of
the internal angle measures of triangle 𝐴𝐵𝐶. If we add the measure of angle
𝐵𝐴𝐶 to both sides of the inequality, then we get that the sum of the measures of
the internal angles in the triangle is greater than two times the measure of angle
𝐵𝐴𝐶.
Of course, we know that the sum of
the internal angle measures in any triangle is 180 degrees. So, we have that 180 degrees is
greater than two times the measure of angle 𝐵𝐴𝐶. We can then divide the inequality
through by two to get that 90 degrees is greater than the measure of angle
𝐵𝐴𝐶. This means that angle 𝐵𝐴𝐶 is
smaller than a right angle, so it must be an acute angle, which is option (A).
