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Let E, F, G, and H be points on a circle such that EF = 15 and GH = 21. Point U is on segment \overline{EF} with EU = 8, and V is on segment \overline{GH} with GV = 5. Line UV intersects the circle at points X and Y. If UV = 6, then find XY.

 

I've got some very nasty equations plz help. :) thx!

 Mar 22, 2020
 #1
avatar+21017 
+2

Using this property:  if two chords of a circle intersect, the product of the sections of one of the chords equals

the product of the sections of the other.

 

Since EF = 15 and EU = 8, UF = 7.

Since GH = 21 and GV = 5, HV = 16.

 

On chord XUVY, call section XU = x and VY = y.

 

Using chord EF and chord XUVY:  7·8  =  x(y + 6).

Using chord GH an chord XUVY:  5·16  =  y(x + 6)

 

Using 5·16  =  y(x + 6)   --->   80  =  y(x + 6)   --->   y  =  80 / (x + 6)

Using 7·8  =  x(y + 6)   --->   56  =  xy + 6x   --->   xy  =  56 - 6x   --->   y  =  (56 - 6x) / x

 

Setting these two equations equal to each other:  80 / (x + 6)  =  (56 - 6x) / x

Cross-multiplying:                                            (56 - 6x)(x + 6)  =  80x

Simplifying:                                                         x2 + 10x - 56  =  0

                                                                           (x + 14)(x - 4)  =  0

                                                                                               x  =  4

 

From here, you can calculate both the value of y and the full length of the chord.

 Mar 22, 2020
 #2
avatar+485 
+2

If you want to find out more about what geno said, I believe the proper name for it is called power of a point theorem! There are three different variations to it, so I would definitely go and check that out!

jfan17  Mar 22, 2020
 #3
avatar+111330 
+1

See the folllowing  [ not to scale] image :

 

 

 

 

 

GV = 5

So VH  = 21 - 5 =  16

 

And 

EU = 8

So UF = 15 - 8  = 7

 

Call  segment UX  = m  and  segment  VY = n

VX = 6 + m

UY = 6 + n

 

Using the intersecting  chord theorem.....we have that

GV * VH  = VX * VY

5 * 16 =  (6 + m) n      (1)

80  = 6n + mn

80 - 6n = mn

 

And 

EU * UF =  UY * UX

8 * 7  =  (6 + n) m

56 = 6m + mn

56 - 6m = mn

 

This implies  that

80 - 6n = 56 - 6m

80 - 56 = 6n - 6m

24  = 6(n - m)

4 = n - m

m + 4  = n   (2)

 

So using (1) and (2) we have  that

80  = (6 + m) (m + 4)

80 = m^2 + 10m + 24

m^2 + 10m - 56  =  0

(m + 14) ( m - 4) =  0

 

Setting both factors to 0 and solving for m  produces  m =  -14   (reject)  or  m  =  4

 

So

m + 4 =  n

4 + 4 =  n

8  = n

 

So.....the length of XY  is

 

m + UV +  n  =

 

4  +  6  +  8   =

 

18

 

 

cool cool cool

 Mar 22, 2020

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