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Completing the Square for Conic Sections

When the equation of a conic section isn't written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows: [more…]

How to Identify Radii, Chords, and Diameters

When you work with circles, there are three straight-line components that you need to be able to identify: radii, chords, and diameters. [more…]

What You Need to Know About a Circle's Radius and Chords

When you’re working with a circle, there are five important theorems that you need to know about the properties of the circle’s radii and chords. (There are really just three ideas; but two of the theorems [more…]

Six Important Circle Theorems

The six circle theorems discussed here are all just variations on one basic idea about the interconnectedness of arcs, central angles, and chords (all six are illustrated in the following figure): [more…]

How a Tangent Relates to a Circle

A line is tangent to a circle if it touches it at one and only one point. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Check out the bicycle [more…]

How to Determine the Length of an Arc

An arc’s length means the same commonsense thing length always means — you know, like the length of a piece of string (with an arc, of course, it’d be a curved piece of string). Make sure you don’t mix [more…]

How to Use Extra Radii to Solve a Problem

When buying a home, the three most important things to consider are location, location, location. With circles, it’s radii, radii, radii. In circle problems, you often need to add extra radii and partial [more…]

How to Determine the Measure of an Angle whose Vertex Is Inside a Circle

An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article covers angles that have their vertex inside a circle—so-called [more…]

How to Solve a Common-Tangent Problem

The common-tangent problem is named for the single tangent segment that’s tangent to two circles. Your goal is to find the length of the tangent. These problems are a bit involved, but they should cause [more…]

How to Determine the Area of Sectors and Segments of a Circle

Mark off a section of a circle with an arc and a chord, and you have a segment (this type of segment has nothing to do with a line segment). Throw a couple of radii around an arc, and you have a sector [more…]

How to Determine the Measure of an Angle whose Vertex Is on a Circle

Of the three places an angle’s vertex can be in relation to a circle (inside, on, or outside the circle), the two types of angles that have their vertex [more…]

How to Determine the Measure of an Angle whose Vertex Is Outside a Circle

An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article discusses the three types of angles that have their vertex outside a circle: secant-secant angles, [more…]

How to Use the Chord-Chord Power Theorem

The Chord-Chord Power Theorem was named for the fact that it uses a chord and — can you guess? — another chord!

Chord-Chord Power Theorem: If two chords of a circle intersect, then the product of the measures [more…]

How to Use the Tangent-Secant Power Theorem

You can solve some circle problems using the Tangent-Secant Power Theorem. This theorem states that if a tangent and a secant are drawn from an external point to a circle, then the square of the measure [more…]

How to Use the Secant-Secant Power Theorem

You can use the Secant-Secant Power Theorem to solve some circle problems. This theorem involves — are you sitting down — two secants! (If you’re trying to come up with a creative name for your child like [more…]

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