# ACT Math: A Challenging Arc Length & Sector Area Problem

One of our more popular posts involves Arc Lengths & Sector Areas for circles. With that in mind, we thought we’d give you a challenging problem (well, really two problems) involving this topic. Give it a try and then scroll down for the answer.

The circle below has its center at O and triangle AOB is equilateral with sides that are each 6 inches. 1.  What is the area, in square inches, of the shaded region?

A.  6π – 3√3
B.  9π – 3√3
C.  6π – 6√3
D.  6π – 9√3
E.  9π – 9√3

# ACT Math: A Challenging Polygons Problem

We’ve listed this problem under “ACT Math,” but you could just as easily see a problem like this on the SAT Math test as well. Give the problem a try, and then scroll down for the solution.

A regular polygon is covered by a piece of paper so that only one of its interior angles is visible. You are able to determine that the interior angle is four times as large as the exterior angle. How many sides does the polygon have? A.  8

B.  9
C.  10
D.  11
E.  12

# Another Challenging Shaded Region Problem

We’ve written about shaded region problems before (here and here) and recently offered up a challenging problem (here) on this topic. Here’s another problem that will put your math brain to work. Give it a try!

In the diagram below, segment AB is tangent to the circle at Point A. The measure of ∠OBA is 30 degrees and the circumference of the circle is 8π units. What is the area, to the nearest tenth of a square unit, of the shaded region? A.  4.8
B.  5.5
C.  5.8
D.  6.0
E.  6.1

# ACT Math: Looking for the “Clever” Solution

The Math portion of the ACT tests how well you’ve mastered the math you’ve been taught in middle school and high school. But on some questions, it does more than that. There are some questions that test how deeply you know the material, how good you are at finding an efficient solution to a difficult problem. In a sense, these problems test how “clever” you are.

Consider the following problem:

If xy = 28 and x + y = 7, what is the sum of the reciprocals of x and y?

A.  1/28
B.  1/7
C.  1/4
D.  4/7
E.  3/4

# ACT Math: A Challenging Circle Problem

Here’s another challenging problem for you, this one involving circles. You’ll need to use concepts you’ve learned in both Algebra and Geometry to get this one right. Good luck!

In the circle below with center at Point O, the length of chord AB is 30 units. The radius in the diagram is perpendicular to the chord AB, which cuts the radius into two pieces with lengths of x units and x + 1 units as labeled in the diagram. What is the length, in units, of the radius of the circle? A.  13
B.  15
C.  17
D.  18
E.  20