It does not surprise anyone when they learn that the properties of circles are tested on the GMAT. Most test-takers will nod and rattle off the relevant equations by rote: Area = Π*radius^2; Circumference = 2Π* radius; etc. However, many of my students are caught off guard to learn that the equation for a circle on the coordinate plane is our good friend the Pythagorean theorem. Why on earth would an equation for a right triangle describe a circle?
Remember: the GMAT loves to test shapes in combination: a circle inscribed in a square, for example, or the diagonal of a rectangle dividing it into two right triangles. So you should expect that triangles will appear just about anywhere – including in circles. Especially in coordinate geometry questions, where the coordinate grid allows for right angles everywhere, you should bring the Pythagorean Theorem with you to just about every GMAT geometry problem you see, even if the triangle isn’t immediately apparent. Let’s talk about how the Pythagorean Theorem can present itself in circle problems – “Pythagorean circle problems” if you will. (And note that the Pythagorean Theorem doesn’t have to “announce itself” by telling you you’re dealing with a right triangle! Very often it’s on you to determine that it applies.)
Take a look at the following diagram in which a circle is centered on the origin (0,0) in the coordinate plane:
Designate a random point on the circle (x,y). If we draw a line from the center of the circle to x,y, that line is a radius of the circle. Call it r. If we drop a line down from (x,y) to the x-axis, we’ll have a right triangle (and an opportunity to therefore apply the Pythagorean Theorem to this circle):
Note that the base of the triangle is x, and the height of the triangle is y. So now we have our Pythagorean Theorem equation: x^2 + y^2 = r^2. This is also the equation for a circle centered on the origin on the coordinate plane. [The more general equation for a circle with a center (a,b) is (x-a)^2 + (y-b)^2 = r^2. When a circle is centered on the origin, (a,b) is simply (0,0.)]
This Pythagorean equation of a circle ends up being an immensely useful tool to use on the GMAT. Take the following Data Sufficiency question, for example:
A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?
(1) The radius of the circle is 4
(2) The sum of the coordinates of P is 0
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
C. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D. EACH statement ALONE is sufficient
E. Statements (1) and (2) TOGETHER are NOT sufficient
So let’s draw this, designating P as (x,y):
Now we draw our trust right triangle by dropping a line down from P to the x-axis, which will give us this:
We’re looking for x^2 + y^2. Hopefully, at this point, you notice what the question is going for – because we have a right triangle, x^2 + y^2 = r^2, meaning that all we need is the radius!
Statement 1 is pretty straightforward – if r = 4, we can insert this into our equation of x^2 + y^2 = r^2 to get x^2 + y^2 = 4^2. So x^2 + y^2 = 16. Clearly, this is sufficient.
Now look at Statement 2. If the sum of x and y is 0, we can say x = 1 and y = -1 or x = 2 and y = -2 or x = 100 and y = -100, etc. Each of these will yield a different value for x^2 + y^2, so this statement alone is clearly not sufficient. Our answer is A.
Takeaway: any shape can appear on the coordinate plane, and given the right angles galore in the coordinate grid you should be on the lookout for right triangles, specifically. If the shape in question is a circle, remember to use the Pythagorean theorem as your equation for the circle, and what would have been a challenging question becomes a tasty piece of baklava. (We are talking about principles elucidated by the ancient Greeks, after all.)
And a larger takeaway: it’s easy to memorize formulas for each shape, so what does the GMAT like to do? See if you can apply knowledge about one shape to a problem about another (for example, applying Pythagorean Theorem to a circle). For this reason it’s important to know the “usual suspects” of how shapes get tested together. Triangles and circles work well together, for example:
-If a triangle is formed with two radii of a circle, that triangle is therefore isosceles since those radii necessarily have the same measure.
-If a triangle is formed by the diameter of a circle and two chords connecting to a point on the circle, that triangle is a right triangle with the diameter as the hypotenuse (another way that the GMAT can combine Pythagorean Theorem with a circle).
-When a circle appears in the coordinate plane, you can use Pythagorean Theorem with that circle to find the length of the radius (which then opens you up to diameter, circumference, and area).
In general, whenever you’re stuck on a geometry problem on the GMAT a great next step is to look for (or draw) a diagonal line that you can use to form a right triangle, and then that triangle lets you use Pythagorean Theorem. Whether you’re dealing wit a rectangle, square, triangle, or yes circle, Pythagorean Theorem has a way of proving extremely useful on almost any GMAT geometry problem, so be ready to apply it even to situations that didn’t seem to call for Pythagorean Theorem in the first place.