Question

chords \\( \overline { a c } \\) and \\( \overline { b d } \\) intersect at e, with \\( b d = 7 \\) units, de = 2 units, and \\( a e = 4 \\) units. what is the length of segment ce? units

Explanation:

Step1: Find the length of BE

Since \(BD = 7\) units and \(DE=2\) units, then \(BE=BD - DE\).
\(BE = 7 - 2=5\) units.

Step2: Use the intersecting chords theorem

The intersecting chords theorem states that if two chords \(\overline{AC}\) and \(\overline{BD}\) intersect at \(E\), then \(AE\times CE=BE\times DE\).
We know \(AE = 4\) units, \(BE = 5\) units, \(DE = 2\) units. Let \(CE=x\) units.
Substitute the values into the formula: \(4\times x=5\times2\).
So, \(4x = 10\).
Solve for \(x\): \(x=\frac{10}{4}=\frac{5}{2}=2.5\) units.

Answer:

\(2.5\)