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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
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Attachment:
Untitled.png
Untitled.png [ 7.53 KiB | Viewed 9738 times ]
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


GMATinsight
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) 5√3
C) 6√2
D) 6√3
E) 8
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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
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
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!
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Attachment:
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


GMATinsight
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) 5√3
C) 6√2
D) 6√3
E) 8

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