Question

1. two secants, \\( \overline { a b c } \\) and \\( \overline { a d e } \\), are drawn to a circle from external point a. if \\( a b = 4, b c = 6 \\), and \\( a d = 5 \\), find de.

Explanation:

Step1: Apply the secant - secant rule

The secant - secant rule states that if two secants \( \overline{ABC} \) and \( \overline{ADE} \) are drawn to a circle from an external point \( A \), then \( AB\times AC=AD\times AE \).
First, find \( AC \). Since \( AC = AB + BC \), and \( AB = 4\), \( BC = 6 \), then \( AC=4 + 6=10 \).
Let \( DE=x \), then \( AE=AD + DE=5 + x \).
Substitute into the formula: \( 4\times10=5\times(5 + x) \).

Step2: Solve the equation

Expand the right - hand side of the equation: \( 40 = 25+5x \).
Subtract 25 from both sides: \( 40-25=5x \), so \( 15 = 5x \).
Divide both sides by 5: \( x=\frac{15}{5}=3 \).

Answer:

\( DE = 3 \)