What is the area of the circular region

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melguy: Master | Next Rank: 500 Posts; Posts: 335; Joined: Mon Mar 21, 2011 11:31 pmFollowed by: 2 members; Location: Australia / India; Thanked: 37 times

by melguy » Fri Aug 09, 2013 12:30 am

I am confused with the language of the question. The question is asking what is the area of the circular region centered at the origin although in the answer explanation it is telling us to find area of the triangle!

Its just the language of the question that is bothering me. I thought after we find the hypotenuse we need to find the area of the circle. Although in the explanation only the hypotenuse is found and that is the final answer.

OA is D.

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Source: Beat The GMAT — Problem Solving | Fri Aug 09, 2013 12:30 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Aug 09, 2013 1:41 am

What is the area of the circle shown below?

When a shape is drawn on the coordinate plane, look for what the shape and the coordinate plane have IN COMMON.
Point (-10, 24) is both a point the circle and the vertex of the triangle shown below:

The red triangle is a multiple of a 5-12-13 triangle:
10-24-26.
When shapes overlap, look for what the shapes have in common.
The hypotenuse of the red triangle is also the RADIUS of the circle.
Thus, the radius of the circle = 26.
Since the area of a circle = πr², we get:
A = π26² = 676π.

The correct answer is D.

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macattack: Senior | Next Rank: 100 Posts; Posts: 90; Joined: Wed Jun 26, 2013 10:55 pmFollowed by: 1 member; Thanked: 11 times

by macattack » Mon Aug 12, 2013 12:38 am

For those who are not familiar with the 5-12-13 triangle or frequently forget that this family exists here is another way: Area of a circle= pi*r^2
In the above figure r=sqrt(10^2+24^2)=sqrt(676)
so r^2 =676--->Area=676*pi
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Brent@GMATPrepNow: GMAT Instructor; Posts: 16206; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Aug 12, 2013 6:34 am

As Mitch has pointed out, we can often solve geometry questions quickly if we're aware of some popular right triangles that feature integer lengths.
The two most popular right triangles are the 3-4-5 and the 5-12-13 right triangles (and, of course, multiples of these triangles such as 6-8-10, and 10-24-26 right triangles)

There are 2 more right triangles you might want to memorize: 8-15-17 and 7-24-25. These right triangles don't appear frequently, but if they do, you can save yourself the hassle (and time) of applying the Pythagorean Theorem.

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