A chord of length 5 cm subtends an angle of 60° at the centre of a cir

Question

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

A. 2π
B. 8
C. 6√2
D. 5√3
E. 10

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nick1816 · Answer

∠AOB =60
OA=OB
Hence, ∠OAB = ∠OBA =60

OAB is an equilateral triangle, so OA=OB = 5

Draw a perpendicular on CD from O.

Triangle OEC is (60-90-30) triangle; Hence, CE = 5*(√3/2)

CD = 2*CE = 2* 5*(√3/2) = 5√3

OR

5/sin30 = CD/sin120

CD = 5√3

GMATinsight · Answer

Answer: Option D

Detailed explanation is in picture attached

P.S. Use of trigonometry should be strictly avoided by test aspirants

fauji · Answer

Formula to find chord length if radius r and angle a are given:  2rSin(\frac{a}{2})

Given:
- We have chord length 5 cm, and angle is 60 deg.
- Since length to both ends of chord is same(i.e. radius), we have equilateral triangle with 5cm each side.

Now, radius = 5 cm and angle is 120 deg.

To find chord length: 2rSin(a/2) =>  2*5*Sin (\frac{120}{2})  =  10*\frac{\sqrt{3}}{2}

=>  5\sqrt{3}

Option D

nick1816 · Answer

Untitled.png Suppose AB makes 60° at the centre of a circle. Join OA and OB. Since OA=OB, angle A= angle B; hence, OAB is an equilateral triangle. OA=OB=AB=5 Suppose CD makes 120° at the centre of a circle. OC:OD:CD = 1:1:√3 = 5:5:5√3 CD = 5√3

CrackverbalGMAT · Answer

Let’s draw the figure to match the conditions given in the question. The figure looks like the one below:

Attachment:

7th May 2020 - Reply 5 - 1.jpg [ 23.42 KiB | Viewed 9432 times ]

From the above, we can conclude that the radius of the given circle is 5 cm. Let us now draw another circle of radius 5 cm which has a longer chord which makes an angle of 120-degree at the centre. The diagram looks like the one below:

Attachment:

7th May 2020 - Reply 5 - 2.jpg [ 24.99 KiB | Viewed 9374 times ]

The perpendicular drawn from the centre of a circle onto a chord bisects the chord.

This perpendicular also happens to bisect the central angle in this case, and hence we have TWO 30-60-90 right angled triangles.

In each of these triangles, hypotenuse = 5 cm. Therefore,

Side opposite to the 30-degree angle =\( \frac{5}{2}\) and

Side opposite to the 60-degree angle (half of the chord) = (\(\frac{5}{2}\))√3

Since half of the chord = (\(\frac{5}{2}\))√3, length of the chord = (\(\frac{5}{2}\))√3 * 2 = 5√3

The correct answer option is B.

Hope that helps!

hasanloubani · Answer

the second triangle is isoceles so the sides must be in ratio 1:1:√2. why you take the 3rd side as √3?

bumpbot · Answer

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A chord of length 5 cm subtends an angle of 60° at the centre of a cir

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A chord of length 5 cm subtends an angle of 60° at the centre of a cir

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