Sometimes the most difficult thing to do on an SAT question is to see something that isn’t there. The SAT provides test takers with enough information to solve the questions, but sometimes the information is not stated explicitly. The easiest way for the makers of the SAT to disguise information is to give written description rather than a picture. Luckily, it doesn’t require Picasso’s eye for line to translate words into pictures. Here is an example:
“A circular field has a line drawn from one end through its center and continuing for ten meters past the edge of the field. A fence begins at one point on the field’s edge and ends at the end of the line not contained by the circle. If the distance from field to the fence is half the total length of the field, what is the total distance of the line and the fence combined?”
This is quite an intricate description and is very hard to visualize. Instead, draw a picture to approximate the description of the elements that have been provided.
This picture gives some helpful information. First off, the two lines are creating what appears to be two sides of a triangle. This is a common set up on SAT math questions and is good to recognize. It is also clear that the “fence” line is tangent to the circle. The question also stated that the distance from the field to the fence, which was given as ten yards, is half the length of the field. This means that the diameter of the circle is two times ten or twenty yards. This is a fantastic start and deserves some self congratulation, but there is still a bit more to be done. The length of the fence is still unknown. Even though the question didn’t state that this figure is a triangle, because a triangle is more useful than two lines it is a good idea to draw the triangle in. We are now imagining lines, who said math doesn’t use your imagination?
Because the fence is tangent to the circle, it creates a right angle with the radius that is drawn to it. We now have a very important piece of information that was previously missing. The third leg of the triangle is a radius of the circle! This means that its length is half the diameter of the circle, or ten meters, and the hypotenuse is the radius plus the ten meter piece of line, or twenty meters. If there is a right triangle which has a side of ten and a hypotenuse of twenty, alarm bells should start going off. What kind of triangle has a hypotenuse that is twice its small side? A 30-60-90 triangle! This means that the fence length is the length of the small side times the square root of three. Thus, the total length of the line plus the fence is twenty (the diameter) plus ten (the line between the circle and the fence) plus ten times the square root of three (the length of the fence). The answer would likely be listed as 30 + 10√3 in a multiple choice question.
This question is very difficult without the aid of the information provided by the pictures and imaginary lines. When it is possible to create a useful shape like a square or a right triangle by imagining lines, it is a good idea to draw those lines to help the test taker draw conclusions that would otherwise be difficult to draw. It can be hard to see what isn’t there, but with a little practice, the hidden pictures can reveal hidden solutions. Happy test preparation!
David Greenslade is a Veritas Prep SAT instructor based in New York. His passion for education began while tutoring students in underrepresented areas during his time at the University of North Carolina. After receiving a degree in Biology, he studied language in China and then moved to New York where he teaches SAT prep and participates in improv comedy. Read more of his articles here, including How I Scored in the 99th Percentile and How to Effectively Study for the SAT.