Tangent & Secant Theorems

When lines intersect with circles, they create segments with special mathematical relationships. These relationships give us formulas that help solve for missing values.

The Two Secants Segments Theorem states: When two secants are drawn from a common point outside a circle, the product of one secant segment and its external part equals the product of the other secant segment and its external part. As a formula: a$a+b$ = c$c+d$.

The Tangent Secant Segment Theorem gives us: When a tangent and a secant are drawn from a common point outside a circle, the square of the tangent segment equals the product of the secant segment and its external part. As a formula: a² = b$b+c$.

Remember this! Both theorems involve multiplication, but tangent segments get squared while secant segments are multiplied by their total length.

Let's see these theorems in action with examples:

• Example 1: Using the two secants theorem, we can find x in 8$8+x$ = 6$6+18$, giving us x = 10
• Example 2: With the tangent-secant theorem in 18² = 10$10+x$, we get x = 22.4
• Example 3: For 15$15+27$ = x·45 using two secants, we find x = 14
• Example 4: When x² = 4$4+12$ with the tangent-secant theorem, x = 8
• Example 5: In 18$18+x$ = 16$16+24$, x = 17.6
• Example 6: For x² = 4$4+5$, x = 6

With practice, you'll quickly recognize which theorem to apply based on whether you're dealing with a tangent, secants, or both.