The Discovery Channel’s “Shark Week” may have more thrills than the GMAT, but the jaws of those deadly sea-predators are a great inspiration to look at one of the GMAT’s own mysterious creatures: circles. Since we miss Shark Week around here, we give you “Arc Week” today.
When it comes to circles, most of us are old pros at finding the area and circumference, and setting up basic ratios and proportions with the parts of a circle, but there are several lesser-known theorems involving the arcs of a circle that might be helpful to have up your sleeve for a GMAT rainy day!
Before we examine these theorems, let’s refresh some “circle” vocab! An arc is simply a portion of the circumference of a circle. A line that goes from one point to another on the circle’s circumference is called a chord. The diameter is a special chord that passes through the circle’s center. A line that touches the circumference at one point only is called a tangent, and a line that intersects the circle at two points is called a secant.
Now onto the theorems:
Theorem #1: an angle formed by two intersecting chords in a circle measures half the sum of the intercepted arcs
Theorem #2: an angle formed by two secants is equal to half the difference of the intercepted arcs
The coolest thing about these theorems is that they apply whether or not the chords and secants intersect in the center of the circle. So if you are looking at a circle with intersecting chords/secants on the GMAT and aren’t sure if you can set up a ratio or proportion since you don’t know where inside the circle these lines cross, you can still determine a significant amount of information! Let’s try a sample question:
What is the value of minor arc y?
(1) Angle x = 70
(2)Angle z = 25
Starting with Statement (1), let’s use our first theorem. An angle formed by two intersecting chords in a circle measures half the sum of the intercepted arcs. That means angle x = ½ (95 + y). Given x = 70, we can plug in and solve for y: 70 = ½ (95 + y). 140 = 95 + y, therefore y = 45. Sufficient.
Statement (2) gives us the value of z. Let’s use our second theorem. We know an angle formed by two secants is equal to half the difference of the intercepted arcs. Therefore, Angle Z = ½ (95 – y). Since we are given angle z, we can solve for y: 25 = ½ (95 – y). 50 = 95 – y, so -45 = -y, and y = 45. Sufficient.
The correct response is (D).
While these theorems are NOT likely to be expressly tested on the GMAT, it’s helpful to expand our knowledge of circle properties to look for interesting or unusual ways statements might be sufficient on Test Day!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing advice to help students better prepare for the GMAT and the SAT.