SAT Tip of the Week: Hidden Shapes in Math Problems

SAT Tip of the Week - Full Some of the most difficult kinds of problems in the math section of the SAT are the problems where there doesn’t seem to be enough information present to solve.  Fear not brave test-taker! Often times, a problem that seems to be lacking simply has information hidden somewhere in the question. But like the great detectives of the past, it is possible to use our wits to find this information.  The first step is to know the common shapes hidden in SAT math questions.

 

1. For non-right triangles, look to draw a height. Here is an example problem:

The triangle shown here is isosceles.  If the angle measure of the vertex opposite one of the sides is 60 degrees, what is the area of the triangle in terms of x?

 

This is a surprisingly simple problem when you get down to it. However, it does require us to put on our Sherlock Holmes cap for a moment.  First, lets fill in what we know.  The fact that this is an isosceles triangle with one angle measuring 60 degrees should quickly tip us off that this is an equilateral triangle. Now the information we are looking for is the area, which is ½ the triangle’s base times the triangle’s height. It would then behoove us to draw a height in our triangle above.

If we recall the rules of the special triangles (30-60-90, 45-45-90), we will realize that the height, in terms of x, is simply the small side of the triangle times the square root of 3 so the height would be ½ x?3.  Plugging this information back into the area equation, the area of the whole triangle would be ½ x (½ x?3) which simplifies to (½ x)²?3.  Once we draw in the height this becomes a fairly standard area problem: how useful finding the hidden pieces can be!

 

2. When possible, draw radii and diameters. One of the most common places information is hidden is within the relationship between the length of radii and other measurements given in a problem.  Here is an example problem:

Two congruent circles are inscribed within a rectangle. What is the length of the diagonal of this rectangle?

 

This problems is more straight forward than the first, but it also requires a little detective work. The first step is to draw some diameters and radii, just to see if it helps us notice anything.

AHA! This small act has yielded a lot of information.  We see that the the diameter is equal to the small side of the rectangle and the length is equal to twice the diameter of the circle. This means the long side is twelve units and the diagonal can be calculated by using the Patagonian theorem.

 

 

6² + 12² = x²

x² = 180

x = ?180

There are many other ways the SAT can hide information from our sight, but these are two important tools in helping us to fill in some of the information that might not have been provided.  Though it can be difficult to see what pieces are missing, with a little practice, you will be a first rate detective.  Happy studying!

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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.

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