Two chords, AB and CD, cross at point P inside a circle. Segments AP, PB, and CP have lengths of 6 cm, 8 cm, and 4 cm, respectively. How long is the chord CD in total?
16 cm
When two chords of a circle cross at an interior point, the intersecting-chords theorem says the product of the two segments of one chord equals the product of the two segments of the other.
Chord AB is split by P into \(AP = 6\) cm and \(PB = 8\) cm; chord CD is split into \(CP = 4\) cm and the unknown \(PD\).
Apply the theorem: \(AP\times PB = CP\times PD\).
Substitute the known values: \(6\times 8 = 4\times PD\), i.e. \(48 = 4\,PD\).
Solving gives \(PD = \dfrac{48}{4} = 12\) cm.
The full chord CD is the sum of its two parts: \(CD = CP + PD = 4 + 12 = 16\) cm.
Key concept: intersecting chords satisfy \(AP\cdot PB = CP\cdot PD\); find the missing segment, then add.
The length of chord CD is 16 cm.
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