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GMATPrep 1 - Geometry question

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amirp: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Sun Dec 20, 2009 2:02 pm; Thanked: 3 times

by amirp » Mon Aug 02, 2010 1:49 pm

Hi guys,

Can anyone tell me how you find 12.5 for this question? I can get it by estimation...

for example:

I estimated that 10.5 < area < 14 but is there a faster or more straight forward solution to this?

Thanks

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Source: Beat The GMAT — Problem Solving | Mon Aug 02, 2010 1:49 pm

Brian@VeritasPrep: GMAT Instructor; Posts: 1031; Joined: Thu Jul 03, 2008 1:23 pmFollowed by: 255 members; Location: Malibu, CA; Thanked: 716 times; GMAT Score:750

by Brian@VeritasPrep » Mon Aug 02, 2010 2:22 pm

Hey Amir,

Great question. One of my favorite things about coordinate geometry is that you can always find a right triangle, so that's the paradigm I use to approach these, and it usually works out pretty efficiently. My first take on this one was that, if that bottom angle (P) looks like a right angle, and if I could prove that it is I'll be able to use that base-height combination of those two lines that create it.

The advantage there is that those two lines both form right triangles with the x and y axes, so those lengths will be easy to calculate. They're actually both 3-4-5 triangles, so the length of each is 5.

Now, you can use the right triangle of that third side, which has a height of 1 and a length of 7, making its Pythagorean Theorem application:

1^2 + 7^2 = side^2
50 = side^2
Side = sqrt 50

Remember, my goal here was to prove that angle P was a right angle, which would make this problem significantly easier. If the isosceles sides are each 5, and the opposite side is sqrt 50, that fits with the 45-45-90 triangle ratio (x, x, sqrt2 * x), then we've proven that angle P is a right angle.

Because that's a right angle, we can use 5 as the base, 5 as the height, and the formula:

A = 1/2 bh = 1/2*5*5 = 12.5

When you're using coordinate geometry, use the fact that the whole coordinate grid is full of right angles - and therefore right triangles - to your advantage!

Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep

Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.

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JeetGulia: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Mon Jun 14, 2010 4:46 am; Thanked: 3 times

by JeetGulia » Mon Aug 02, 2010 2:28 pm

amirp wrote:Hi guys,

Can anyone tell me how you find 12.5 for this question? I can get it by estimation...

for example:

I estimated that 10.5 < area < 14 but is there a faster or more straight forward solution to this?

Thanks

I will give you a hint it has to do with slope of line..and you can calculate the perpendicular of the triangle...and then you can calculate the area...

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amirp: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Sun Dec 20, 2009 2:02 pm; Thanked: 3 times

by amirp » Mon Aug 02, 2010 2:45 pm

Both approaches make total sense. Thank you guys.

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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 » Mon Aug 02, 2010 7:33 pm

amirp wrote:Hi guys,

Can anyone tell me how you find 12.5 for this question? I can get it by estimation...

for example:

I estimated that 10.5 < area < 14 but is there a faster or more straight forward solution to this?

Thanks

The area of the rectangle drawn around triangle PQR is 28. Since triangle PQR takes up less than half the rectangle, PQR < 14.
Only answer choice A works.

The correct answer is A.

Last edited by GMATGuruNY on Thu Feb 23, 2012 5:08 am, edited 1 time in total.

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getso: Master | Next Rank: 500 Posts; Posts: 141; Joined: Sat Feb 28, 2009 8:19 am; Thanked: 1 times

by getso » Mon Aug 02, 2010 10:11 pm

GMATGuruNY wrote:
amirp wrote:Hi guys,

Can anyone tell me how you find 12.5 for this question? I can get it by estimation...

for example:

I estimated that 10.5 < area < 14 but is there a faster or more straight forward solution to this?

Thanks
Estimating is the quickest way to solve this problem. See the attached .pdf. The area of the rectangle drawn around triangle PQR is 28. PQR takes up less than half the rectangle, so PQR < 14. Only answer choice A works: 12.5 < 14.

Great Explanation..

Quickest way to solve without much calculation!!

Thanks GMATGuruNY

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selango: Legendary Member; Posts: 1460; Joined: Tue Dec 29, 2009 1:28 amFollowed by: 7 members; Thanked: 135 times

by selango » Mon Aug 02, 2010 10:29 pm

Area of triangle when vertices are given,

area=| [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]/2 |

(x1,y1)=(4,0)

(x2,y2)=(0,3)

(x3,y3)=(7,4)

=[4(-1)+0+7(-3)]/2=|-12.5|=12.5

--Anand--

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