Question

chords \\(\overarc{ac}\\) and \\(\overarc{bd}\\) intersect at e, with bd = 7 units, de = 2 units, and ae = 4 units. what is the length of segment ce? \boxed{} units

Explanation:

Step1: Find the length of BE

Since \( BD = 7 \) units and \( DE = 2 \) units, we can find \( BE \) by subtracting \( DE \) from \( BD \). So, \( BE=BD - DE = 7 - 2=5 \) units.

Step2: Apply the Intersecting Chords Theorem

The Intersecting Chords Theorem states that if two chords \( AC \) and \( BD \) intersect at point \( E \), then \( AE\times CE=BE\times DE \). We know \( AE = 4 \) units, \( BE = 5 \) units, and \( DE = 2 \) units. Let \( CE=x \). Substituting the known values into the formula, we get \( 4\times x=5\times 2 \).

Step3: Solve for CE

Simplify the right - hand side of the equation: \( 4x = 10 \). Then, divide both sides by 4 to solve for \( x \): \( x=\frac{10}{4}=\frac{5}{2} = 2.5 \).

Answer:

\( 2.5 \)