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

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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
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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.

Image

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 --> \(area = \pi{r^2}=\pi{x^2}\).

Answer: B.,

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

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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

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

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New post 06 Sep 2017, 22:00
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.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

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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, 22:00
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