Welcome to the world of pi :-). So far in Geometry you’ve examined the properties of shapes with straight edges. Just like those shapes can be broken down into their components, so can circles. Circles are a big deal as you move into Algebra II and Pre-Calculus, so it’s important to get a good understanding of this topic to carry with you for success in future courses. To begin, an arc is a portion of the circumference, or a curved line that runs along the edge of the circle. A chord is any straight line that passes through the circle with endpoints on the circumference. If a chord passes through the center of the circle and bisects it, it is called the diameter.
Minor Arc of a Circle and Major Arc of a Circle:
A minor arc is defined as the shorter of the two arcs formed when a circle is divided by a chord. The length of a minor arc is less than half the circumference of the circle. Conversely, a major arc is defined as the longer of two arcs formed when a circle is divided by a chord. The major arc is always longer than the minor arc and measures more than half the circumference of the circle.
How to Measure the Length of an Arc:
A central angle is an angle whose vertex is on the center point of the circle.
To say that an angle subtends an arc means that it’s rays pas through the endpoints of that arc. Both these ideas are important when measuring arcs. First the ratio of the central angle to the total angle of the circle (360 degrees) must be determined. The formula for the measure of an arc is this ratio times the total circumference of the circle.
Tangent Radius Theorem:
A tangent line is an external line that intersects the circumference of a circle in exactly one point. The Tangent Radius Theorem establishes a relationship between a radius of a circle and a tangent line drawn to that circle at the point of tangency (where the tangent line and the circle intersect). According to this theorem, the radius at the point of tangency is perpendicular to the tangent line.
Common tangents refer to lines that are tangent to two or more circles simultaneously. There are two types of common tangents: internal and external. Internal common tangents are located between the circles, while external common tangents are found outside the circles. The number of common tangents between two circles depends on whether or not they intersect, and if so, how they intersect. For example, when two circles do not intersect, there are four common tangents – two internal, and two external. If the circles touch externally exactly once, there are three common tangents – one internal, and two external. If the circles overlap, there are only two common tangents – both external. Last, if they completely overlap, there is only one external common tangent.
Lines Intersecting Outside a Circle:
Lines that intersect outside a circle create interesting configurations that give rise to a series of angles and geometric relationships. For instance, the angle formed by a secant and a tangent line drawn from an external point to the circle is equal to half the difference of the intercepted arcs. The same holds true for angles formed by two secant lines, or two tangent lines.
Segment Products Theorem:
The Segment Products Theorem, also known as the Power of a Point Theorem, establishes a relationship between the lengths of segments formed by two intersecting chords or secants within a circle. It states that for a circle and a point not on the circumference of the circle, the products of the lengths of the two segments is constant along any line through the point and the circle.
Equation of a Circle:
The equation of a circle is usually where your Algebra II class will pick up next year at the beginning of the school year. It is a mathematical representation that describes the set of all points in a plane that are equidistant from a fixed point, known as the center. The distance from the center to any point on the circle is called the radius.
The general form of the equation of a circle with center (h,k) and radius r that touches the circumference at (x,y) is given by:
(x – h)^2 + (y – k)^2 = r^2
This equation is derived from the Pythagorean theorem, where the square of the distance between any point on the circle and the center is equal to the square of the radius.