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# Point A lies on a circle whose center is

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Manager
Joined: 23 Feb 2017
Posts: 59
Point A lies on a circle whose center is  [#permalink]

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05 Jan 2018, 09:41
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3
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Difficulty:

85% (hard)

Question Stats:

34% (01:57) correct 66% (01:57) wrong based on 54 sessions

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Point A lies on a circle whose center is at point C. Does point B lie inside the circle?

1) $$BC^2 = AC^2 + AB^2$$
2) ∠CAB is greater than ∠ABC

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Math Expert
Joined: 02 Sep 2009
Posts: 49862
Point A lies on a circle whose center is  [#permalink]

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05 Jan 2018, 10:19
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Point A lies on a circle whose center is at point C. Does point B lie inside the circle?

(1) $$BC^2 = AC^2 + AB^2$$.

This statement implies that triangle ABC is right angled at A (BC being hypotenuse) So, AB is perpendicular to AC, which is radius (AB is tangent to the circle). Thus, B is outside the circle. Sufficient.

Little technicality here: this statement to be sufficient it should be mentioned that A and B are different points. I'm assuming it's given

(2) ∠CAB is greater than ∠ABC.

Larger side of a triangle is opposite larger angle. Hence, this statement implies that BC is larger than AC (radius). Thus, B is outside the circle. Sufficient.

We could use the same logic also for (1): BC (hypotenuse) is larger than AC (radius). Thus, B is outside the circle. Sufficient.

Hope it's clear.

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Point A lies on a circle whose center is &nbs [#permalink] 05 Jan 2018, 10:19
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