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# Given the circle to the right, with center O, diameter AOB, a radius

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Math Expert
Joined: 02 Sep 2009
Posts: 61475
Given the circle to the right, with center O, diameter AOB, a radius  [#permalink]

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23 Mar 2016, 11:47
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Difficulty:

5% (low)

Question Stats:

91% (00:45) correct 9% (01:33) wrong based on 87 sessions

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Given the circle to the right, with center O, diameter AOB, a radius of 5, and the inscribed triangle ABC, what is the length of AC?

A. 4.5
B. 5
C. 6
D. 6.5
E. 7

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circletrianglequestion.gif [ 5.04 KiB | Viewed 2550 times ]

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Re: Given the circle to the right, with center O, diameter AOB, a radius  [#permalink]

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23 Mar 2016, 19:38
ACB is a right angle triangle.
AB = 10 is the hypotenuse
BC = 8
AC = $$\sqrt{10^2 - 8^2}$$ = 6

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Re: Given the circle to the right, with center O, diameter AOB, a radius  [#permalink]

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04 Apr 2016, 14:16
Angle within a semicircle is always 90 => the triangle is a right triangle => AC = √(100-64) => 6 i.e.C is correct
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Re: Given the circle to the right, with center O, diameter AOB, a radius  [#permalink]

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01 Dec 2019, 14:51
Hi All,

We're told that in the circle with center O, AOB is the diameter, the radius of the circle is 5 and triangle ABC is inscribed. We're asked for the the length of AC. This question is based on some specific Geometry patterns - and if you recognize them, then you can answer this question without doing much math at all.

To start, any triangle in a circle that has all 3 vertices ON the circumference AND has one side that is the diameter of the circle is a RIGHT TRIANGLE. Thus, Angle C is a 90-degree angle.

Next, since the radius of the circle is 5, the diameter is 10. We now know two of the sides of the right triangle (8 and 10 - which is the hypotenuse). You can use the Pythagorean Theorem (A^2 + B^2 = C^2) to find the missing side OR you might recognize that we're dealing with a 3/4/5 right triangle that has been 'doubled' (into a 6/8/10 right triangle). Thus, the missing side is 6.

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Rich
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Re: Given the circle to the right, with center O, diameter AOB, a radius   [#permalink] 01 Dec 2019, 14:51
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