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# Points A and B are on a circle of radius 5 and AB = 6. Point C is the

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Math Expert
Joined: 02 Sep 2009
Posts: 58396
Points A and B are on a circle of radius 5 and AB = 6. Point C is the  [#permalink]

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27 Mar 2019, 21:07
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Difficulty:

55% (hard)

Question Stats:

55% (01:47) correct 45% (02:24) wrong based on 22 sessions

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Points A and B are on a circle of radius 5 and AB = 6. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC?

(A) $$\sqrt{10}$$

(B) 7/2

(C) $$\sqrt{14}$$

(D) $$\sqrt{15}$$

(E) 4

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Joined: 27 Sep 2018
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Re: Points A and B are on a circle of radius 5 and AB = 6. Point C is the  [#permalink]

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27 Mar 2019, 22:22
1
Let O be centre of the circle

As AB is the arc and it's midpoint is given as C. If we draw a perpendicular from to AB it will bisect the chord AB at D(assumed point) and meet C such that OC=5=OD+DC
NOW, TRIANGLE OAD is rt angled triangle
And we can find OD using Pythagoras OD=rt(5^2-3^2)=4

Similarly triangle ADC is also a rt angled triangle and we can find AC using Pythagoras, AC=rt(3^2+1^2)=rt10

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Re: Points A and B are on a circle of radius 5 and AB = 6. Point C is the  [#permalink]

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28 Mar 2019, 03:29
chetan2u can you please explain this?
Math Expert
Joined: 02 Aug 2009
Posts: 8004
Points A and B are on a circle of radius 5 and AB = 6. Point C is the  [#permalink]

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28 Mar 2019, 07:01
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Shreshtha55 wrote:
chetan2u can you please explain this?

Let join AB, and CO. Now, CO will bisect AB in two halves of 6/2 or 3..
Now since C is in the middle, the other thing will be that the chord AB and radius OC will be perpendicular to each other.

So, take triangle ADO with hypotenuse as 5 and one side as 3, so sides are 3:4:5, so OD = 4.

Now let us take triangle ADC, again a right angled triangle. CD=5-4=1, and AD = 3, so AC = $$\sqrt{1^2+3^2}=\sqrt{10}$$.
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Re: Points A and B are on a circle of radius 5 and AB = 6. Point C is the  [#permalink]

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28 Mar 2019, 12:17
chetan2u wrote:
Shreshtha55 wrote:
chetan2u can you please explain this?

Let join AB, and CO. Now, CO will bisect AB in two halves of 6/2 or 3..
Now since C is in the middle, the other thing will be that the chord AB and radius OC will be perpendicular to each other.

So, take triangle ADO with hypotenuse as 5 and one side as 3, so sides are 3:4:5, so OD = 4.

Now let us take triangle ADB, again a right angled triangle. BD=5-4=1, and AD = 3, so AC = $$\sqrt{1^2+3^2}=\sqrt{10}$$.

Hi chetan2u,
Thank you for your solution,but some portions are not clear, can you check , in the last line did you mean triangle ADC rather than ADB also did you mean CD rather than BD,

Thanks.
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Re: Points A and B are on a circle of radius 5 and AB = 6. Point C is the   [#permalink] 28 Mar 2019, 12:17
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