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Math Expert V
Joined: 02 Sep 2009
Posts: 58431
In the diagram above, AB = 10 is the diameter of the circle, and AC =  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 57% (01:48) correct 43% (02:09) wrong based on 67 sessions

HideShow timer Statistics In the diagram above, AB = 10 is the diameter of the circle, and AC = 6. Given that point C is inside the circle, which could be the length of BC?

I. 7
II. 8
III. 9

A. I
B. II
C. III
D. I & II
E. II & III

Attachment: triangle in circle.JPG [ 11.71 KiB | Viewed 879 times ]

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In the diagram above, AB = 10 is the diameter of the circle, and AC =  [#permalink]

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Bunuel wrote: In the diagram above, AB = 10 is the diameter of the circle, and AC = 6. Given that point C is inside the circle, which could be the length of BC?

I. 7
II. 8
III. 9

A. I
B. II
C. III
D. I & II
E. II & III

Attachment:
triangle in circle.JPG

Here we can think of three properties

1) Sum of any two sides of the triangle > Third side
i.e. 16>BC >4

2) Angle drawn at circumference in a semicircle is a right angle
But here C is not on circumference i.e. ACB is an Obtuse angle triangle

3) For any obtuse triangle $$c^2 > a^2 + b^2$$ where c is the longest side of triangle

i.e. $$b^2 < 10^2 - 6^2$$
i.e. $$b^2 < 100 - 36$$
i.e. $$b^2 < 64$$
i.e. $$b< 8$$

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General Discussion
Intern  B
Joined: 15 Nov 2018
Posts: 9
Re: In the diagram above, AB = 10 is the diameter of the circle, and AC =  [#permalink]

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GMATinsight wrote:
Bunuel wrote: In the diagram above, AB = 10 is the diameter of the circle, and AC = 6. Given that point C is inside the circle, which could be the length of BC?

I. 7
II. 8
III. 9

A. I
B. II
C. III
D. I & II
E. II & III

Attachment:
triangle in circle.JPG

Here we can think of three properties

1) Sum of any two sides of the triangle > Third side
i.e. 16>BC >4

2) Angle drawn at circumference in a semicircle is a right angle
But here C is not on circumference i.e. ACB is an Obtuse angle triangle

3) For any obtuse triangle $$c^2 > a^2 + b^2$$ where c is the longest side of triangle

i.e. $$b^2 < 10^2 - 6^2$$
i.e. $$b^2 < 100 - 36$$
i.e. $$b^2 < 64$$
i.e. $$b< 8$$

How did you know that this traiangle is an obtuse-angled triangle ?
Since the point C is not on the circumference of a triangle, this is not a right triangle I agree.

Now for any acute-angled triangle, $$a^2 + b^2 > c^2$$ (where 'C' is the longest side)
Now, I can see from the three options given that all of them are less than 10. So '10' must be the longest side.

Hence, we need to see for which of the squares of 7/8/9 and 6 are less than the square of 10.

And that is for option A.
For option B, $$a^2 + b^2 = c^2$$
For option C, $$a^2 + b^2 < c^2$$

Kindly let me know what is wrong with my line of reasoning. Re: In the diagram above, AB = 10 is the diameter of the circle, and AC =   [#permalink] 22 Sep 2019, 06:35
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In the diagram above, AB = 10 is the diameter of the circle, and AC =

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