Sunday, July 14, 2013

# What is “Angle Bisector”? GRE Math Fact

2:19 AM
Bisector is a term you may encounter in GRE geometry problems. So what the heck is a bisector and how do you use conquer bisector problems on the GRE?
Bisection is the division of something into two equal or congruent parts, usually by a line or a ray, which is then called a bisector. When a bisector, bisects an angle, it divides the angle into two equal parts. In the following figure, $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle HGI$ is bisected by $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \overrightarrow{GM}$ such that $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle HGM\cong\angle MGI$ or in other words the angles are equal. $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \overrightarrow{GM}$ is also referred as the angle bisector of $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle HGI$.

GRE problems can become somewhat more complicated when the ray divided an angle into a specific ratio. In such cases you need to apply little bit of algebra to the underlying geometry concept.
Let’s look at an example problem so that this concept is clear in your mind:
Example:
In the above figure, $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \overrightarrow{BD}$ splits the angle $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABC$ in such a way that the measure of angle $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABD$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle DBC$  are in the ratio 2:3. What is the measure of angle $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABD$?
A) 18
B) 20
C) 36
D) 45
E) 54
F) 60
Solution:

Step 1: We know that the sum of the interior angles is equal to the exterior angle. Therefore
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABC=\angle ABD+\angle DBC$
From the figure we see that angle $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABC$ is a right angle. It follows that
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABC=\angle ABD+\angle DBC=90$
Step 2: Since the measure of angle $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABD$and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle DBC$  are in the ratio 2:3, we can write $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABD = 2x$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle DBC=3x$
\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \dpi {120} \begin{align*} 2x+3x&=90 \\ 5x&=90 \\ x &=18 \\ \end{align*}
Step 3: Since $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABD = 2x$
$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} \angle ABD=2\times18=36$