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

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

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New post 12 Feb 2018, 08:20
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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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A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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

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New post 12 Feb 2018, 09:04
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.
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Re: A triangle is inscribed in a circle. A point is randomly chosen inside  [#permalink]

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New post 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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New post 15 Feb 2019, 02:59
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Re: A triangle is inscribed in a circle. A point is randomly chosen inside   [#permalink] 15 Feb 2019, 02:59
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