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ACT Math Tip: Visualization

Happy New Year!

If you’re Math skills are so-so, but you rock your studio art, ceramics, or graphic design elective class, it might surprise you to know that you have an important skill many ACT test-takers lack: the ability to visualize. On tough ACT Math Geometry questions, especially coordinate geometry questions, questions will describe a line, a parabola, or a shape, and neglect to draw it.

How a little artistic talent can lead to a higher ACT Math score:

Many students either fail to attempt to draw their own figure, and thus get the question incorrect, or if they do draw the figure, it’s shape and dimensions don’t really match the given information from the question-stem. If you’re a great artist, that skill will absolutely help you out on these types of problems! Let’s try a question out!

In the xy-coordinate plane, a circle with center (-4, 0) is tangent to the line y = -x. What is the circumference of the circle?

(A) 2π

(B) 2π√2

(C) 4√2

(D) 4π√2

(E) 4/√2

Notice all of the specific requirements of what we have to visualize: a circle at a specific point on a coordinate grid, a line that is “tangent,” or touching the circle at only one point, AND the line has to be drawn so that its equation is y  = -x. That’s a LOT for someone who isn’t good at drawing to attempt. Here’s how to do it:


The line y = -x makes a 45 degree angle with each axis in the second quadrant. Connect the center of the circle to the point of tangency on y = -x. The radius of a circle is perpendicular to its point of tangency.


We can draw a 45-45-90 triangle using the x-axis and y = -x, and use our knowledge of right triangle ratios to find the the radius (or hypotenuse of the triangle) is 2√2.


Circumference = 2πr = 4π√2. The answer is (D). If our drawing hadn’t been so neat and organized, this question would have been absolutely impossible to solve!


Remember these “Golden Rules” for drawing on the ACT Math Test:

1. Draw your picture in a BIG way!

Don’t keep the diagram tiny and tucked away in a corner of your test booklet. Use the empty spaces, even if you have to flip to a different page – in order to visualize, you need to see things blown up to scale.

2. Follow the requirements set out by the problem.

In the example question, there were a LOT of requirements for the position and size of the circle, and the exact slope of the line. Don’t get TOO creative with your drawing such that you accidentally forget about the question’s requirements.

3. Label everything neatly.

Notice how in our diagram above, the line is labeled y = -x, and the point of tangency is neatly labeled (-2,2). This helped us visualize how the triangle fits inside the circle. If you’re having trouble labeling, it’s probably because you didn’t draw a large-enough diagram.

You can see how having a little artistic ability can really help you out when see questions such as these on Test Day! Look for them on your next practice test!


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