In the diagram, PA = 6, PB = 5, PQ = 27, QD = 7, and QC = 12. FInd the length of the chord when PQ is extended on both sides to the circle.
+128474 Let q be the segment of PQ lying at the top of the circle
Let p be the segment of PQ lying at the bottom of the circle
By the intersecting chord theorem we have that
q (27 + p) = 6 * 5 ⇒ 27q + qp = 30 (1)
p(27 + q) = 7 * 12 ⇒ 27p + qp = 84 (2)
Subtract (1) from (2) and we have that
27 ( p - q) = 54 divide both sides by 2
p - q = 2 ⇒ p = 2 + q
Sub this into (2) for p
(2 + q) ( 27 + q) = 84
q^2 + 29q + 54 - 84 = 0
q^2 + 29q - 30 = 0
(q + 30) ( q -1) = 0
Only q - 1 0 produces a positive result for q
So q = 1
p = 2 + q = 3
So
PQ extended = 27 + p + q = 27 + 3 + 1 = 31
