We all know the formula for calculating the area of a circle (Area = ), and most of us also remember how to compute its circumference (Length = ). There is, however, much more to circles than these two equations. Other properties are equally useful on the GMAT, if not so widely known. One such property is the inscribed angle theorem, which describes a special relationship between angles and circles. Because this theorem features prominently on the GMAT, it is well worth reviewing.
If we start with a basic circle, we can create an angle by joining the center of the circle to two radii. The resulting angle is known as a central angle, since the vertex of the angle is the center of the circle itself (the other two points lie on the circumference). The diagram below features circle O, where its two radii, OA and OB, join together at the center to form central angle ∠AOB:
Incidentally, every central angle also defines an arc. In the diagram above, central angle ∠AOB defines the arc AB. An arc can actually be measured in two separate ways: by length and by angle. Much like a line, an arc has a distance that it spans, and so its length can be measured. Much like a central angle, however, an arc can also be measured in degrees, in which case the arc simply has the same value (in degrees) as the central angle which defines it. Arc AB is simply x°.
There are also other interesting angles that do not have a vertex at the center. One special type of circle-related angle has all of its three points lying on the circumference. If we connect these points, we end up with an inscribed angle, like ∠BCA to the right:
An inscribed angle essentially consists of two chords — line segments that lie on the circle — connected to a point that lies on the circumference.
It turns out that there is an interesting proof that relates an inscribed angle with the measure of its arc. Known as the inscribed angle theorem, it states that the measure of the inscribed angle is simply half the measure of the arc it defines. This property is absolutely invaluable for helping us solve geometry problems on the GMAT:
Since arc AB measures 2x, an inscribed angle with that same arc will measure exactly half that (which is simply x).
There is an alternative way to express the inscribed angle theorem. Since a central angle and the arc it defines are equivalent, the following diagram is also true:
According to the inscribed angle theorem, the inscribed angle (x) is exactly ½ the central angle (2x).
To see how we can use this helpful theorem on the GMAT, let’s look at the following problem to the right:
If BD=DC, what is the value of ‘x‘ in the figure to the right?
We’d like to take advantage of the inscribed angle formula, but as you’ll notice, ∠BAD and ∠BCD don’t share the same arcs, so we can’t apply the formula to relate the two angles. Let’s fix that by adding some new line segments. We’ll connect points B and C to form the isosceles triangle BCD. The base angles of this triangle are (180°-30°)/2=75° each. Using the inscribed angle theorem, we then find the measure of the arcs defined by each inscribed angle.
Inscribed angle ∠CBD is 75°, so arc CD is 150°; similar logic helps us find the measure of arc BAD. We can also apply the inscribed angle formula for ∠BDC to find that arc BC is 60°.
At this point, we can calculate the measure of arc BCD = 60 + 150 = 210. Since inscribed angle BAD defines arc BCD, the inscribed angle must be half of 210, so that x = ½×(210°) = 105°.
Now that you’ve learned how to use the inscribed angle theorem, I’d like to leave you with an interesting challenge problem:
In the regular octagon to the right, what is the value of central angle x?