The figure below shows 2 tangent circles such that the 10 -centimeter diameter of the smaller circle is equal to the radius of the larger circle. What is the area, in square centimeters, of the shaded region? \ \(\mathbf{F} .10\) G. 75 H. \(\quad 5 \pi\) J. 10\(\pi\) K. 75\(\pi\)

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. When solving geometry problems, understanding the basic shapes and their attributes is crucial. One fundamental shape is the circle, which is perfectly round and is defined by its center and its radius. The radius is a straight line from the center to any point on the edge of the circle. A key aspect of solving problems involving circles, like the given exercise, is knowing the relationship between a circle's radius and its diameter—the diameter is twice the length of the radius.

In our exercise, the interplay between the two circles and their shared dimensions provide vital clues to solving for the shaded area. By recognizing that circles can have different sizes yet share a common diameter and radius, and that the area of a circle increases with the square of its radius, students can unlock the answers to complex geometry problems.

Circles have a set of unique properties that are essential to understanding and solving problems related to them. One of the most important properties is the formula used to calculate a circle's area: \( A = \pi \times r^2 \), where \( A \) is the area and \( r \) is the radius. This formula tells us that the area of a circle is directly proportional to the square of its radius, a concept which we used in the step-by-step solution to calculate the areas of the two tangent circles.

Another vital property is that when two circles are tangent to one another, they do not overlap, and the distance between their centers is equal to the sum of their radii. These relationships allow students to visually and mathematically recognize how to compute the areas necessary to find the shaded region as shown in the exercise.

Preparing for the ACT Math section involves mastering various mathematics topics, including algebra, geometry, and trigonometry. For geometry, specifically, it's important to practice problems that involve circle properties, among other shapes, to be ready for the variety of questions that may appear on the exam. The given exercise is an excellent example of a typical ACT geometry question. Students can prepare by familiarizing themselves with the formulas for calculating a circle's area, understanding the relationship between a circle's radius and diameter, and practicing problems that involve subtracting areas to find the area of shaded regions, just like we did in the step-by-step solution.

Practice also includes improving your skills in visualizing geometric arrangements and manipulating equations to arrive at the correct answers. Timed quizzes and exercises can be beneficial in helping students get used to the pace of the ACT, ensuring that they can solve problems quickly and efficiently under exam conditions.