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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.

 

 

 Dec 13, 2020
 #1
avatar+117546 
+1

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

 

cool cool cool

 Dec 13, 2020
 #2
avatar+1164 
0

The top is 1, and the bottom is 3

 Dec 13, 2020

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