Question Video: Finding the Type of Angle in a Triangle Using the Median Mathematics

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).