Two perpendicular chords intersect in a circle. The segments of one ch

Question

Two perpendicular chords intersect in a circle. The segments of one chord are 3 and 4; the segments of the other are 6 and 2. Then the diameter of the circle is:

A.  \sqrt{56} 
B.  \sqrt{61} 
C.  \sqrt{65} 
D.  \sqrt{75} 
E.  \sqrt{89}

ENGRTOMBA2018 · Accepted Answer

Look at the attached image for a explanation of the variables and dimensions.

Given,

GF = 3, GE = 4, BG = 6 and CG = 2

O is the center of the circle. Chords BC and EF are perpendicular to each other.

Now, in triangle OBC, OB = OC = radius of the circle, thus triangle OBC is an isosceles triangle and OA  \perp  BC (OA is perpendicular to BC). Thus OA will bisect BC into 2 equal parts  (a property of an isosceles triangle that the perpendicular drawn from one of the vertex to the base (assuming base is NOT one of the 2 equal sides), bisects the base).

---> AB = AC = (6+2)/2 = 4 ....(1)

Similarly in triangle OFE, OF = OE = radius of the circle. OD  \perp  EF ---> FD = DE = (3+4)/2 = 3.5 ---> FD = DG + FG ---> DG = FD - FG = 3.5 - 3 = 0.5 ...(2)

Also, OAGD form a rectangle such that GD = OA = 0.5 and AG = OD = 2.

Thus, in right triangle OAB, OA^2+AB^2 = OB^2  ---> r^2 = 0.5^2+4^2 = 16.25  --->  r = \sqrt{16.25}  ---> diameter =  2r = 2 \sqrt{16.25}  =  \sqrt{4*16.25} = \sqrt{65}  .

C is the correct answer.

sagarsir · Answer

option C Half Length of chord 1 = (3+4)/2 = 3.5 Similarly half length of chord 2 = 4 Ow we can form two right angles with radii as hypotenuse: 1) with legs 3.5 and 4-2 = 2. Radius = sqrt(3.5^2 + 2^2) = sqrt(16.25) 2) with legs 4 and 3.5-3 = .5 radius = sqrt(4^2 + .5^2) = sqrt(16.25) In each case the diameter obtained is 2* sqrt(16.25) = sqrt(65) Option C Kudos if you like

dominicraj · Answer

Rule: Perpendicular from centre to the chord bisects the chord.

Given that, we can find out that the point of intersection of the chords is 0.5  units and 2  units away from the centre.

Once the centre distance from point of intersection is received we can use distance formula to calculate the radius:
(4*4+0.5*0.5)^2=sqrt(16.25)

Diameter = 2 * sqrt(16.25) = sqrt(65). Option C

chetan2u · Answer

pl refer sketch..
radius= \sqrt{2^2+3.5^2} = \sqrt{16.25} 
dia=2* \sqrt{16.25} = \sqrt{4} * \sqrt{16.25} = \sqrt{65} 
ans C

Shree9975 · Answer

Hi,
 VeritasPrepKarishma , can you please explain this with diagram.

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Two perpendicular chords intersect in a circle. The segments of one ch

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