Question

chords ab and cd intersect at point e. in the diagram shown, ae = 3 units, be = 4 units, de = x units, units. diagram: circle with chord ab (a---e---b, ae=3, eb=4) and chord cd (d---e---c, de=x, ec=2x) what is the exact length, in units, of (overline{de})?

Explanation:

Step1: Apply Intersecting Chords Theorem

When two chords intersect in a circle, the products of the lengths of their segments are equal. So, \( AE \times BE = DE \times CE \).
Given \( AE = 3 \), \( BE = 4 \), \( DE = x \), and \( CE = 2x \), we substitute these values into the formula:
\( 3 \times 4 = x \times 2x \)

Step2: Solve the Equation

Simplify the left - hand side: \( 12 = 2x^{2} \)
Divide both sides by 2: \( x^{2}=\frac{12}{2}=6 \)
Take the square root of both sides. Since length cannot be negative, we consider the positive square root: \( x = \sqrt{6} \) (we discard the negative solution \( x=-\sqrt{6} \) as length is non - negative)

Answer:

\(\sqrt{6}\)