What is the length of the chord ?

Geometry Level 1

There is a chord AB in a circle with center O. As shown in the figure, a line segment OC makes an angle of 60 degrees with the chord. If OC=6cm and BC=8cm, then find the length of the chord.


The answer is 10.

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24 solutions

Smit Kiri
Mar 25, 2014

Consider a line segment OD which is a perpendicular bisector of AB. As cos60 = DC/OC, we get DC = 3. Now BC = DC + BD from which we get BD = 5. Length of the chord AB is 2 x BD = 10.

8-6=2 and finally 8+2=10

ishan tandon - 7 years, 2 months ago

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hahaha yeah right

Ryan Abando - 7 years, 2 months ago

draw another line from O to the chord at 60.now length of each side of the equilateral triangle thus formed is 6.The length left is 2.Hence chord length is 6+2+2=10

Atul Vaibhav - 7 years, 2 months ago

Wow, and here I go using Law of Cosines to find the radius when I clearly could have just did it your way. Nice and simple solution.

Daniel Liu - 7 years, 2 months ago

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I did it the same stupid way hahaha

Rory Bramley - 7 years, 2 months ago

cos 60=1/2=x/6... hence x=3... 8-3=5... and finally ac=5-3=2... so AB=AC+BC= 2+8=10....

Sreekanth Gopalakrishnan - 7 years, 2 months ago

simple 8-6=2 becz equilateral triangle and 8+2=10

Rohit Singh - 7 years, 2 months ago

Good question

Poonam Kashyap - 7 years, 2 months ago

10

Nitesh Kumar Jha - 7 years, 2 months ago

length of the Perpendicular will be 3 root(3) = 6 sin60 and the corresponding base will be 3 = 6*cos60 so half of the chord value 5 only [Perpendicular to chord divide it equally] ] so value is 10

Gautam Singh - 7 years, 2 months ago

10

Shubham Chaudhari - 7 years, 2 months ago

BC-OC=AC AC+BC= AB AO=OB 8cm-6cm=2 and finally 8 +2= 10

Johnny Balseca Villacrés - 7 years, 2 months ago

>>solve"cos(60)=(8-x)/(2*6)";
>>x=
2
>>length=(8+x) ;
>>length =
10


mostafa algammal - 7 years ago

there is a better approach, draw a line D = 60 ° \quad D =60\degree\quad from o such that it touches B C \quad BC\quad and,now C O D \quad\triangle COD\quad is equatorial due to its 60 , 60 , 60 \quad 60,60,60\quad angles. now, C D = 6 c m \quad CD=6cm\quad and D B = 2 c m \quad DB =2 cm\quad we can say A C = D B = 2 c m \quad AC=DB=2cm 6 + 2 + 2 = 10 \therefore 6+2+2=\boxed{10}

Aareyan Manzoor - 6 years, 5 months ago

another simpler way, is to construct a line that is 60 degree from o and lies on bc, name it op so angle opc is also 60 degree that is equilateral triangle so pc = 6 cm. From that 8-6 = 2 so 8+2 = 10

A Former Brilliant Member - 7 years, 1 month ago

simple. no need to use calculator. OC is 6cm, BC is 8cm and angle OCB is 60degrees, if you add another point within segment AB, lets say point D that should give an angle ODC or ODA = 60degrees. You can form a equilateral triangle. All sides of equilateral triangle is equal. So if OC is 6cm, OD and DC is also 6cm. We know that BC is 8cm and DC is 6cm, subtract the 2 segments to get the segment BD, and that would be 2cm. SO, AC is 2cm too. Add the 3 segments AC, CD and BD = 10cm. :) make sense?

very nice

Ravi Bendi - 7 years, 2 months ago

i found it same too

Emran Tahir - 7 years, 2 months ago

very nice method

Shahid Hussain - 7 years, 2 months ago
Manish Sagar
Apr 3, 2014

Another way. Draw angle ODC=60 and C-D-B. Now triangle OCD is equilateral. So CD=6 and so BD=BC-CD=2.. Same way AD will also be 8 and AC will be 2. So AB=AC+CD+DB=2+6+2=10. No trigonometry required :-)

yeah i thought in the same way no need of trignometry

Reshma Raj - 7 years, 2 months ago
Ahmad Brisha
Apr 3, 2014

10 cm (construction drop perpendicular from the center we will get aright angled triangle with 30 and 60 degrees . and the opposite side to the angle 30 equals half the hypotenuse =3 cm so half the chord =8-3 =5 so the chord = 10

Samir Shaheen
Apr 3, 2014

It is very easy. Just compute 6 cos 60 = 3 cm which is the length of the projection if the radius on AB. Now AB will half of AB WILL BE 5 cm and AB will be 10 cm.

10

Hargovind Singh - 7 years, 2 months ago
Rifath Rahman
Aug 30, 2014

OC=6,Now lets draw a perpendicular OE on the chord ,so the COE is a right triangle,that makes its base CE=6 * cos 60=6 * *1/2=3,now EC=BC-CE=8-3=5,so 1/2 * chord=5 so chord=10

Bhushan Varadkar
Apr 23, 2014

The sum is wrong as, let perpendicular on cord be OP. In triangle OPC. angle OPC is 60deg... hence OP is 3root(3)... in tranangle OAP, angle OAP is 45 .. hence. AP is 3root(3).. hence chord length is 6root(3).. i.e = 10.92

Vikash Yadav
Apr 20, 2014

draw an imaginary line OD intersecting AB in such a way that OD=OC=6cm and points C & D do not coincide, thus making an isosceles triangle OCD where OC=OD & angle(OCD)=angle(ODC). Since the two angles are 60 degrees each, then the third angle will be (180-60-60=60). Hence the triangle becomes equilateral triangle. Hence the third side CD=6cm. Now, according to the question, BC=8cm, BD=BC-CD=8-6=2cm. Hence the length of the chord becomes 6+2+2=10cm

Debajyoti Ghosh
Apr 16, 2014

if we draw a line OD such a way that angle(ODA)=60 degree this will be a equilateral triangle and OD=6 and CD=6...so DB=8-6=2....so AC=2..since AC=BD..total AB=AC+BC=2+8=10...

Kaif Ahsan
Apr 15, 2014

Let's suppose a line segment OD is drawn , as if OD perpendicular AB. Here, sin60= OD/OC = 5.2 and cos60= DC/OC =3. We know that BC=8cm.Means, CD+DB=BC= 8cm. We know that, a perpendicular from the center to a chord divides the chord in half. So, DB = 8-3=5cm. so, AB=2DB= 10. Answer= 10 cm

Krishna Garg
Apr 11, 2014

We draw aperpendicular from o to chord AB as OD.Now with OCD trianle ,side OC is 6 Cm and angle C is 60 degree so CD is 3 cm and therefor BD is 5 cm..OD is 3underroot 3.(hay h) Now from trianle ODB we canculate OB( radius as underroot 52, from h=3underroor3 and DB as 25) Length of chord = 2 underroot rs quare - h square +2 X5 + 10 Ans.

K.K.GARGIndia

Devendra Kumar
Apr 10, 2014

10 thats a simple use of trigonometry

Arthur Paderan
Apr 8, 2014

Draw a segment OD to any point of BC that will make a 60 degree angle namely, angle D then a triangle BOD is formed , so, angle O is also 60 degrees, since all angles of the triangle BOD are equal so the sides are also equal, where BO=BD=OD=6. Since the length of BD= 8 - DC, then BD=6, so, DC=2, where DC=AB so, AC=AB+DB+DC=2+6+2=10

Pratik Ranjan
Apr 8, 2014

drop a perpendicluar ol to chord ab .. in triangle olc angle olc =90 so by trigonometry ab lc=occos60 i.e 6 1/2=3 now bc=bl+lc 8=bl+3 bl= 5 ol is perpendicluar to chord ab it also divides it into two equal parts so ab =2bl =2 5=10

Manohar Mathur
Apr 7, 2014

Drop a perpendicular on AB from centre O. Let this perpendicular meet at X on AB. Then, CX = 6 cos60 = 3. So, BX = 8-3 =5. As perpendicular on chord from centre bisects it into two equal parts, AX=BX. Therefor, AB=2BX= 2x5=10.

CH=cos OCB. OC= cos60.6=1/2.6=3 BH = BC - CH = 8- 3=5 as BH=1/2 . AB -> AB=10.

Ihbashek Patnaik
Apr 5, 2014

if you consider a point on the chord D forming a equilateral triangle....then yoy can find the length DB and from there you can infer the length i.e AC+CD+DB+10

draw a perpendicular from O to chord AB.mark as D.in triangle OCD find OD.now in tiangle OCB apply cosine law ,OB^2=OC^2+BC^2 - 2 OC BCCos(60).In triangle ODB,find BD, using pythagoras theorem, In a circle perpendicular from centre on chord divides the chord into two equal halves.therefore,AB=2*BD=10

Akshay Kumar
Apr 3, 2014

consider a perpendicular bisector from center to the chord now if we take cos60 then we get lenght of cto the pt the pt on the bisector line on the chord now sub that lenght from given value lt of chrd we get 5 make it double we get the lt of the chord

BC=8cm and angle ADO=60 degree so the triangle is equilateral and all side are 6 cm their fore 8-6=2 therefore its 10 cm.

Poornima Iyer
Apr 2, 2014

Draw line OD perpendicular to the chord AB. OD = Sin60 * 6. CD = cos60 * 6 = 3 cm. Now, DB = BC - CD = 5 cm. So, AB = 2* DB = 10 cm

Abid Ali
Apr 2, 2014

join O with B and calculate OB by using cosine law i:e a2=b2+c2-2abcosA tha t will be radius.than apply same formula on triangle OCA and calculate length AC=2 than calculate AC+CB=10

Kartek Pillai
Apr 2, 2014

why don't we try to place another line that has an angle 60 degree, so all the sides of the triangle will be 60 degree which means all will have the same length. Then try minus 8 with 2 so the length on the other will be the same. you get me once you try.Enjoy

Good solution

Abid Ali - 7 years, 2 months ago
Yasmine Babers
Apr 2, 2014

since angle =60 and since we can do construction from o perpendicular on ab in f then it is 30 , 60 triangle then length of cf=0.5 from the hypot.=0.5*6=3 and since cb=8 then fb = 5 since of construction perpendicular then it bisecy ab at f then 5+5=10

GOOD

NISHIMA PASRICHA - 7 years, 2 months ago

good

Srinivasa Reddi - 7 years, 2 months ago

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