Medians, Altitudes and Angle Bisectors in Special Triangles on the GMAT

Quarter Wit, Quarter WisdomNo concept group on GMAT geometry questions is more important than triangles, and no discussion of triangles on the GMAT would be fully comprehensive without covering medians, altitudes, and angle bisectors. So in today’s post, let’s discuss these interesting triangle concepts as a group and show how they can be applicable to challenging GMAT geometry questions. To begin, we are assuming you know the terms median, angle bisector and altitude but still, just to be sure, we will start our discussion today by defining them:

Median – A line segment joining a vertex of a triangle with the mid-point of the opposite side.

Angle Bisector – A line segment joining a vertex of a triangle with the opposite side such that the angle at the vertex is split into two equal parts.

Altitude – A line segment joining a vertex of a triangle with the opposite side such that the segment is perpendicular to the opposite side.

Usually, medians, angle bisectors and altitudes drawn from the same vertex of a triangle are different line segments. But, importantly, in special triangles such as isosceles and equilateral triangles, they can overlap. And, as always, any time you can identify a triangle as a special triangle, you have even more rules you can apply to better understand it. We will now give you some of these properties which can be very useful.

I.

In an isosceles triangle (where base is the side which is not equal to any other side):

– the altitude drawn to the base is the median and the angle bisector;

– the median drawn to the base is the altitude and the angle bisector;

– the bisector of the angle opposite to the base is the altitude and the median.

II.

The reverse of what we just learned is also true. Consider a triangle ABC:

– If angle bisector of vertex A is also the median, the triangle is isosceles such that AB = AC and BC is the base. Hence this angle bisector is also the altitude.

– If altitude drawn from vertex A is also the median, the triangle is isosceles such that AB = AC and BC is the base. Hence this altitude is also the angle bisector.

– If median drawn from vertex A is also the angle bisector, the triangle is isosceles such that AB = AC and BC is the base. Hence this median is also the altitude.

and so on…

III.

In an equilateral triangle, each altitude, median and angle bisector drawn from the same vertex, overlap.

Try to prove all these properties on your own. That way, you will not forget them.

A few things this implies:

-Should an angle bisector in a triangle which is also a median be perpendicular to the opposite side? Yes.

-Can we have an angle bisector which is also a median which is not perpendicular? No. Angle bisector which is also a median implies isosceles triangle which implies it is also the altitude.

-Can we have a median from vertex A which is perpendicular to BC but does not bisect the angle A? No. A median which is an altitude implies the triangle is isosceles which implies it is also the angle bisector.

and so on…

Let’s take a look at a sample Data Sufficiency question to demonstrate the applicability of these concepts:

Question: What is the measure of angle A in triangle ABC?

Statement 1: The bisector of angle A is a median in triangle ABC.
Statement 2: The altitude of B to AC is a median in triangle ABC.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

(D) EACH statement ALONE is sufficient to answer the question asked.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Solution: We are given a triangle ABC but we don’t know what kind of a triangle it is.

So let’s move on to work on the statements directly.

Statement 1: The bisector of angle A is a median in triangle ABC.

The angle bisector is also a median. This means triangle ABC must be an isosceles triangle such that AB = AC. But we have no idea about the measure of angle A. This statement alone is not sufficient.

Statement 2: The altitude of angle B to AC is a median in triangle ABC.

The altitude is also a median. This means triangle ABC must be an isosceles triangle such that AB = BC (Note that the altitude is drawn from vertex B here). But we have no idea about the measure of angle A. This statement alone is not sufficient.

Using both statements together, we see that AB = AC = BC. So the triangle is equilateral! So angle A must be 60 degrees. Sufficient! Therefore the correct answer is (C).

And the ever-important takeaway from this problem: here we were able to use our knowledge of how medians, altitudes, and angle bisectors appear in special types of triangles to prove that we were dealing with a special, equilateral triangle. This is an important lesson both for the concepts of medians and altitudes and for GMAT geometry as a whole, particularly in Data Sufficiency. You have to be proactively on the lookout for special triangles, because special triangles allow you to determine quite a few facts from a limited set of information. They set up perfectly for Data Sufficiency, where the entire question type depends on “cleverly-hidden” information that can be unlocked by recognition of key concepts. So it is very important that you:

1) Embrace any GMAT geometry rules that help you identify or leverage special triangles (equilateral, isosceles, 30-60-90, etc.)

2) Look proactively to find special triangle relationships so that you can apply the rules that accompany them. This is particularly important with Data Sufficiency but also in Problem Solving problems where if you don’t see a relationship you can be doomed to staring in frustration, but if you do you can quickly apply all that you know to reach an answer.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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