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# If O is the center of the circle above and the length of chord AB is 2

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Joined: 02 Sep 2009
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If O is the center of the circle above and the length of chord AB is 2  [#permalink]

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25 Nov 2019, 02:19
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Difficulty:

65% (hard)

Question Stats:

54% (02:24) correct 46% (02:16) wrong based on 24 sessions

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If O is the center of the circle above and the length of chord AB is 2 units, what is the length of the arc ACB?

(1) The area of ΔOAB is $$\sqrt{3}$$ square units

(2) The area of sector OACBO is $$\frac{2π}{3}$$

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If O is the center of the circle above and the length of chord AB is 2  [#permalink]

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Updated on: 28 Nov 2019, 06:08
Bunuel wrote:

If O is the center of the circle above and the length of chord AB is 2 units, what is the length of the arc ACB?

(1) The area of ΔOAB is $$\sqrt{3}$$ square units

(2) The area of sector OACBO is $$\frac{2π}{3}$$

AB=2
OA=OB=r

To find the length ACB we need r and angle OAB

(1) The area of ΔOAB is $$\sqrt{3}$$ square units sufic

Area ΔOAB = √3; ΔOAB is an equilateral or an isosceles triangle;

If ΔOAB is an equilateral triangle, $$area=r^2√3/4=√3…r=2…<OAB=60$$

If ΔOAB is an isosceles triangle with base AB, $$area=AB•h/2=√3…2h/2=√3…h=√3$$
A perpendicular from the O to AB would divide ΔOAB into two right angle triangles with:
base=1 height=√3… this is a special triangle 30:60:90 with sides x:x√3:2x or, 1:√3:2.
Now we have radius = 2 and <OAB = 60.

(2) The area of sector OACBO is $$\frac{2π}{3}$$ insufic

$$<OAB=a$$
$$Area.Sector = πr^2(a/360)…πr^2(a/360)=2π/3…(r,a)=unknown$$

Ans (A)

Originally posted by exc4libur on 26 Nov 2019, 09:23.
Last edited by exc4libur on 28 Nov 2019, 06:08, edited 2 times in total.
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Re: If O is the center of the circle above and the length of chord AB is 2  [#permalink]

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27 Nov 2019, 09:40
1
Statement 1.From the area and the base we can find the height. Since we know this is an isosceles/equilateral, the perpendicular/height to the base divides the base into 2 equal lengths ,dividing the triangle into two congruent triangles with legs 1 and $$\sqrt{3}$$.from this we know we have a hypoteneuse of 2.In essence, we have a 90-60-30 triangle.Since we have the angle(=60) and the radius(=2) we can find the length of the arc.Sufficient

Statement 2 : to find the length of the arc, we need to multiply the area given by 2 and divide the result by the radius of the circle.We can not find the radius
with the information given us.Insufficient

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Re: If O is the center of the circle above and the length of chord AB is 2  [#permalink]

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27 Nov 2019, 11:11
From the question , we can see a triangle inscribed in a circle where 2 legs are the radius. This tells us that it's an isoceles triangle

The length of chord AB is 2 meaning we can divide the triangle into two equal right angles triangle with a base measurement of 1cm on each side.

Statement 1- The area of ΔOAB is root 3 square units:

From the area of the triangle 1/2 *b*h, we can deduce the height of the triangle to be root 3, since we have already established earlier that the base is 1cm .

Going by this, we know we are dealing with a 30-60-90 triangle, and we can figure out that the angle AOB is 60 degrees, and that the radius is 2 square units ..

Hence we can find the length of Arc ------------》 sufficient

Statement 2. The area of sector OACBO is 2pi/3

because we don't know the radius of the circle, we cannot find the angle AOB.
We're left with 2 unknowns... both the radius and the angle SON, hence we cannot deduce the length of Arc
Statement 2 ----》insufficient

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Re: If O is the center of the circle above and the length of chord AB is 2   [#permalink] 27 Nov 2019, 11:11
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