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# A triangle is inscribed in a circle. A point is randomly chosen inside

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Manager
Joined: 30 Dec 2016
Posts: 231
GMAT 1: 650 Q42 V37
GPA: 4
A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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12 Feb 2018, 08:20
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Difficulty:

75% (hard)

Question Stats:

45% (01:36) correct 55% (01:34) wrong based on 148 sessions

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A triangle is inscribed in a circle. A point is randomly chosen inside the circle. What is the probability that point lies on the area common to triangle and circle?

A) Longest side of triangle ABC measures 10.

B) ABC is a right isosceles triangle.

Source: Experts Global

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SandySilva

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A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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12 Feb 2018, 08:45
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sandysilva wrote:
A triangle is inscribed in a circle. A point is randomly chosen inside the circle. What is the probability that point lies on the area common to triangle and circle?

A) Longest side of triangle ABC measures 10.

B) ABC is a right isosceles triangle.

Source: Experts Global

Since we are looking at probability, even finding ratio of areas will be sufficient...

Statement I does not say anything about radius of circle or area of triangle..
Insufficient

Statement II states ABC is isosceles right angle triangle....
Remember the rule - any angle in the semicircle with two sides meeting the end of diameter is 90..
So here the Dia of circle is hypotenuse of triangle...
Since triangle is isosceles, sides of triangle and radius are related..
We can find the ratio of two areas..
Sufficient
. B

Otherwise..
Sides of isosceles right angle triangle = Dia/√2 = 2r/√2=√2r
Area of isosceles triangle =$$\frac{1}{2}*√2r*√2r=r^2..$$
Area of circle=$$π*r^2$$..
Probability = $$\frac{r^2}{π*r^2}=\frac{1}{π}$$
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##### General Discussion
Senior Manager
Joined: 15 Oct 2017
Posts: 295
GMAT 1: 560 Q42 V25
GMAT 2: 570 Q43 V27
GMAT 3: 710 Q49 V39
Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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12 Feb 2018, 09:04
1
B.

A) No information about triangle's type (scalene or isosceles) and radius or diameter of the circle. Not Sufficient.
B) Right isosceles triangle means diameter is the hypotenuse and is equal to 2r.
Area of triangle = 1/2*2r*r = r^2
Area of circle = 22/7*r^2
Therefore probability = r^2/(22/7*r^2) = 7/22 = Sufficient.
Intern
Joined: 15 Oct 2016
Posts: 28
Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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12 Feb 2018, 20:38
sandysilva wrote:
A triangle is inscribed in a circle. A point is randomly chosen inside the circle. What is the probability that point lies on the area common to triangle and circle?

A) Longest side of triangle ABC measures 10.

B) ABC is a right isosceles triangle.

Source: Experts Global

We don't need to do calculations to solve this one!

If you know that it is a right isosceles triangle, then the longest side must be the diameter and the relative orientation and relative length of other sides are known. Therefore, we would also know the ratio of areas of the triangle to the circle. Hence statement B alone is sufficient.
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Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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24 Sep 2019, 01:37
thank you all for the explanation.

Statement 2- The isosceles triangle can be at any point in the circle, and not necessarily cut through the diameter of the circle. In this case, this question is not solvable. Hence, answer is E

Appreciate if someone can advise where did I go wrong on the below thought.
Senior Manager
Joined: 15 Feb 2018
Posts: 367
Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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28 Sep 2019, 18:54
Is there only a single size right-angle triangle that can fit in a circle?
Director
Joined: 20 Jul 2017
Posts: 926
Location: India
Concentration: Entrepreneurship, Marketing
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Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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28 Sep 2019, 19:16
1
philipssonicare wrote:
Is there only a single size right-angle triangle that can fit in a circle?

Hi philipssonicare,

No. There are many right angled triangles that can fit inside a circle. But one side will ALWAYS be the diameter (As we know, angle in a semicircle is 90 degrees).

But Option (2) says right angled ISOSCELES triangle.
There will be only one such triangle, with hypotenuse as diameter.
Eg: Assume a circle with Center origin (0, 0) and radius 2 units and let opposite ends of diameter are A(2, 0) and B(-2, 0) lying on x-axis.
Third vertices will lie on y-axis at (0, 2) or (0, -2).

Hope I’m clear!

Posted from my mobile device
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Joined: 20 Jul 2019
Posts: 38
Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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28 Sep 2019, 19:48
aaronlcr93 wrote:
thank you all for the explanation.

Statement 2- The isosceles triangle can be at any point in the circle, and not necessarily cut through the diameter of the circle. In this case, this question is not solvable. Hence, answer is E
I think it doesn't matter at what point isoscles triangle is , the longest length possible in a circle is diameter, so the longest length of the triangle is diameter because in question it is mentioned triangle is inscribed in a circle
Appreciate if someone can advise where did I go wrong on the below thought.

Posted from my mobile device
Re: A triangle is inscribed in a circle. A point is randomly chosen inside   [#permalink] 28 Sep 2019, 19:48
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