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# Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a

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Intern
Joined: 11 Mar 2014
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Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a [#permalink]

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19 Sep 2014, 20:13
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Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a circle with center C such that each vertex of DEF lies on the circumference of C. If the shortest side of DEF has length x, which of the following represents the area of circle C?

A. 2πx

B. πx^2

C. sq. root 3πx^2

D. 2πx^2

E. 4 πx^2
[Reveal] Spoiler: OA
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Joined: 02 Sep 2009
Posts: 44419
Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a [#permalink]

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20 Sep 2014, 12:53
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neeti1813 wrote:
Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a circle with center C such that each vertex of DEF lies on the circumference of C. If the shortest side of DEF has length x, which of the following represents the area of circle C?

A. 2πx

B. πx^2

C. sq. root 3πx^2

D. 2πx^2

E. 4 πx^2

The triangle has angles in the ratio 1:2:3 --> a + 2a + 3a = 180° --> a = 30°. So, we have 30-60-90 right triangle.

This is one of the 'standard' triangles you should be able to recognize on sight. A fact you should commit to memory is: the sides are always in the ratio $$1 : \sqrt{3}: 2$$.

Since the shortest side is x, then the sides are $$x : \sqrt{3}x: 2x$$.

Next, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle).

So, the largest side, the hypotenuse, 2x, must be the diameter of the circle --> radius = x --> $$area = \pi{r^2}=\pi{x^2}$$.

Hope it's clear.
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Joined: 12 Nov 2016
Posts: 788
Re: Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a [#permalink]

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06 Sep 2017, 22:57
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neeti1813 wrote:
Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a circle with center C such that each vertex of DEF lies on the circumference of C. If the shortest side of DEF has length x, which of the following represents the area of circle C?

A. 2πx

B. πx^2

C. sq. root 3πx^2

D. 2πx^2

E. 4 πx^2

Bunuel this is a PS in the DS directory please move. Kudos
Math Expert
Joined: 02 Sep 2009
Posts: 44419
Re: Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a [#permalink]

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06 Sep 2017, 23:00
Nunuboy1994 wrote:
neeti1813 wrote:
Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a circle with center C such that each vertex of DEF lies on the circumference of C. If the shortest side of DEF has length x, which of the following represents the area of circle C?

A. 2πx

B. πx^2

C. sq. root 3πx^2

D. 2πx^2

E. 4 πx^2

Bunuel this is a PS in the DS directory please move. Kudos

Done. Thank you for noticing.
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Re: Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a   [#permalink] 06 Sep 2017, 23:00
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# Right triangle DEF has angles in the ratio 1:2:3 and is inscribed in a

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