How to Apply the Three Power Theorems to Circle Problems
There are three power theorems you can use to solve all sorts of geometry problems involving circles: the chord-chord power theorem, the tangent-secant power theorem, and the secant-secant power theorem.
All three power theorems involve an equation with a product of two lengths (or one length squared) that equals another product of lengths. And each length is a distance from the vertex of an angle to the edge of the circle. Thus, all three theorems use the same scheme:
This unifying scheme can help you remember all three of the power theorems. And it’ll help you avoid the common mistake of multiplying the external part of a secant by its internal part (instead of correctly multiplying the external part by the entire secant) when you’re using the tangent-secant or secant-secant power theorem.
Using the chord-chord power theorem
The chord-chord power theorem was brilliantly named for the fact that the theorem 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 lengths of the two parts of one chord is equal to the product of the lengths of the two parts of the other chord. (Is that a mouthful or what?)
Put another way,
For example, in the first figure,
Here’s the diagram.
Using the tangent-secant power theorem
The tangent-secant power theorem is another absolutely awe-inspiring example of creative nomenclature.
Tangent-Secant Power Theorem: If a tangent and a secant are drawn from an external point to a circle, then the square of the length of the tangent is equal to the product of the length of the secant’s external part and the length of the entire secant. (Another mouthful.)
In other words:
For example, in this figure, 82 = 4 (4 + 12)
Using the secant-secant power theorem
Last but not least, there is the secant-secant power theorem. Are you sitting down? This theorem involves two secants!
Secant-Secant Power Theorem: If two secants are drawn from an external point to a circle, then the product of the length of one secant’s external part and the length of that entire secant is equal to the product of the length of the other secant’s external part and the length of that entire secant. (The biggest mouthful of all!)
Here’s what it looks like:
For instance, in this figure, 4 (4 + 2) = 3 (3 + 5)