Nate is a content developer at Knewton, and he loves thinking up ways to help students with their GMAT prep.
Geometry is an important part of any GMAT test-taker’s conceptual toolkit. On Data Sufficiency geometry questions, it’s especially key to have an intuitive feel for what is and is not solvable given certain bits of information. Consider the following difficult problem:
A circle having center O is inscribed in triangle ABC. What is the measure of angle BAC?
- The radius of the circle is 2.
- Segment OA has length 4.
(A) Statement (1)Â ALONE is sufficient, but statement (2)Â 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)Â TOGETHER are sufficient to answer the question asked, butÂ 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)Â TOGETHER areÂ NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
There are two ways to attack a problem like this. At the end of this article is an extremely well-thought out, coldly-reasoned, logical, academic explanation. While interesting, in terms of the GMAT it is an absolutely useless way to think about the question. It just takes too much time! Instead, you should train yourself to approach problems like these intuitively. Here’s how:
First, check out this simple geometry tool. It’s a handy JAVA applet that lets you see how the problem works visually. If you take a few moments to play with the applet, you may be able to get a better intuitive feel for the restrictions created by statements (1) and (2).
Try keeping the radius the same while changing the size and shape of the triangle. Notice the angle BAC changing? Next try keeping the length of segment OA the same while moving the circle around to change its radius. See the angle changing again?
By thinking intuitively, you can tell that neither statement is sufficient on its own. When the radius and segment length are fixed, though, it’s another story. Once you know both these pieces of information, you can tell you’re dealing with a 30-60-90 right triangle (more on this below), so finding the measure of angle BAC is a matter working with triangle properties.
That’s why the answer here is C — statements (1) and (2) together are sufficient, but neither is sufficient alone.
The moral here is to avoid wordy reasoning in geometry whenever possible. Practice the art of visualization. You can’t use a nifty applet on test day, but you can draw pictures of extreme cases and move the segments around in your head. This kind of intuitive reasoning is essential on Data Sufficiency geometry questions —where time is short and diagrams are seldom drawn to scale.
Now that you’re thinking visually, take a look at the wordy explanation. Really — do not read the following explanation until you play with the applet! There are tons of sites out there like the one I mentioned above. Spending some time playing with the possible orientations of triangles and circles is going to build your geometric intuition, which will only help your GMAT score.
Here’s the wordy explanation. Look out.
Note that at the three points where the circle touches the triangle, the radius of the circle connecting these points to the center of the circle is going to be perpendicular to each of the triangle’s sides.
Statement 1 tells us that the radius of the circle is 2. Although this defines the circle entirely, there are many possible triangles in which a circle of radius 2 could be inscribed — imagine that the triangle that this circle sits inside of is NOT an equilateral triangle. It is certainly possible that triangle ABC is scalene, in which case each angle has a different measure. Since Statement 1 makes no mention of any of the triangle’s vertices, angles, or sides, note that by simply relabeling the vertices of a scalene version of triangle ABC, we could have different measures of angle BAC. Statement 1 is not sufficient.
Statement 2 tells us that segment OA has length 4. Although this gives us more specific information about the vertex named in angle BAC, note that, again, this statement makes no restrictions on the measure of the angle BAC. Imagine a really skinny acute triangle where angle BAC is very small, and the distance between point A and the segment BC is very close to the length of segment OA = 4. Or imagine a really fat obtuse triangle where angle BAC is very large, and the distance between point A and the segment BC is just slightly less than twice the length of segment OA = 4. In the first case, the measure of angle BAC is small, in the second case, the measure of angle BAC is close to 180 degrees. Statement 2 is not sufficient.
Taken together, the two statements tell us two important pieces of information about the smaller triangle AOD, where D is the point where the circle touches segment AC. Segment OD is a radius of the circle, so from statement 1 we know that it has length 2, and statement 2 tells us that OA has length 4. Knowing that the angle ODA is 90 degrees (remember that the radii of the circle are perpendicular to the triangle at each of the points of tangency) tells us that the triangle OAD is a right triangle. Furthermore, we know that the hypotenuse of the right triangle, OA, is twice the length of the leg, OD, the radius of the circle of length 2. This is beginning to look a lot like a 30-60-90 triangle. Noting that it would violate the triangle inequality if the leg DA were less than or equal to the length of leg OD, we can conclude that the leg OD = the radius of the circle is the shortest leg of the triangle, and we do indeed have a 30-60-90 triangle.
Now we can conclude that angle OAD is equal to 30 degrees, being opposite the shortest leg, but what about angle OAE? We need to know the measure of this angle in order to find the measure of angle BAC. Symmetry applies here. All statements applied to segments OD and DA also apply to OE and EA because triangles AOD and AOE are congruent — they share side OA, angles OEA and ODA are equal (right angles), and the triangles have their shortest sides equal to the radius of the circle. Hence the measure of angle BAC is twice the measure of angle OAD: 2 Ã— 30 = 60. The statements together are sufficient.