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# The figure above shows a circle with center O. If point R in triangle

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Math Expert
Joined: 02 Sep 2009
Posts: 52971
The figure above shows a circle with center O. If point R in triangle  [#permalink]

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05 Sep 2018, 22:41
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Difficulty:

25% (medium)

Question Stats:

88% (01:04) correct 13% (00:50) wrong based on 25 sessions

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The figure above shows a circle with center O. If point R in triangle PQR is equidistant from points P and Q, what is the area of the inscribed triangle PQR?

(1) The triangle is a right triangle.
(2) The radius of the circle is 8.

Attachment:

image003.gif [ 1.83 KiB | Viewed 366 times ]

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Re: The figure above shows a circle with center O. If point R in triangle  [#permalink]

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05 Sep 2018, 23:31
Bunuel , my doubt is that we know if a triangle is inscribed in a semicircle, it is always 90 degree. In this question, we can say that since PQ is the part of the triangle, it is a straight line. So is there any need of Statement 1?
Math Expert
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Re: The figure above shows a circle with center O. If point R in triangle  [#permalink]

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05 Sep 2018, 23:32
Bunuel wrote:

The figure above shows a circle with center O. If point R in triangle PQR is equidistant from points P and Q, what is the area of the inscribed triangle PQR?

(1) The triangle is a right triangle.
(2) The radius of the circle is 8.

Attachment:
image003.gif

Now PQ is the diameter, so triangle PQR is a right angled triangle ...
also R is equidistant from P and Q so PR and QR are equal and thus triangle PQR becomes right angled isosceles triangle......
thus dia = $$PQ = 2r$$ and the sides $$PR=QR=r\sqrt{2}$$
area of PQR = $$\frac{1}{2}*PR*QR=\frac{1}{2}*(\sqrt{2}*r)^2=r^2$$

so we just require the RADIUS

(1) The triangle is a right triangle.
insuff

(2) The radius of the circle is 8.
area = $$8^2=64$$
suff

B
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Re: The figure above shows a circle with center O. If point R in triangle  [#permalink]

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05 Sep 2018, 23:35
A gives the same info as given in diagram already so it is not useful. B gives radius and the center of hypotenuse of a right triangle is equidistant from all vertices so height of triangle = radius = 8, so area is 32. Answer is B.

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Re: The figure above shows a circle with center O. If point R in triangle  [#permalink]

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05 Sep 2018, 23:38
Believe answer is B as 1st option provided is a fact,
"In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle ∠ABC is a right angle."
and we have length of one of the base so we should be able to find the rest.
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Re: The figure above shows a circle with center O. If point R in triangle   [#permalink] 05 Sep 2018, 23:38
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