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massi2884: Senior | Next Rank: 100 Posts; Posts: 95; Joined: Tue Sep 13, 2011 3:19 pm; Thanked: 3 times; GMAT Score:710

by massi2884 » Thu May 10, 2012 10:54 am

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In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA ?

(1) AC^2 = 2(AD)^2

(2) â–³ABC is isosceles

OA C

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Source: Beat The GMAT — Data Sufficiency | Thu May 10, 2012 10:54 am

aneesh.kg: Master | Next Rank: 500 Posts; Posts: 385; Joined: Mon Apr 16, 2012 8:40 amFollowed by: 29 members; Location: Pune, India; Thanked: 186 times

by aneesh.kg » Thu May 10, 2012 11:06 am

Area (ABC) = 1/2*AC*BC
Area (DBA) = 1/2*DA*AB

Statement(1):
If AD = x, AC = (2)^0.5 * x. We don't know the the base of the two triangles.
INSUFFICIENT

Statement(2):
Not enough data.
INSUFFICIENT

On combining, AC = CB = (2)^0.5 *x, AB = (2)^0.5 * (2)^0.5 * x = 2x, AC = (2)^0.5 *x

We have the values of the base and altitude of both the triangle in terms of x, and hence their areas can be compared.

[spoiler](C)[/spoiler] is the answer.

Aneesh Bangia
GMAT Math Coach
[email protected]

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peterrm555: Newbie | Next Rank: 10 Posts; Posts: 4; Joined: Mon Jul 18, 2011 4:58 pm

by peterrm555 » Tue Nov 13, 2012 2:27 pm

Very nicely done and it only took a few lines.
The OG explanation is a nightmare, following their explanation it will take at least 5 minutes.
Thanks

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bpolley00: Master | Next Rank: 500 Posts; Posts: 192; Joined: Wed Jul 11, 2012 5:04 pmFollowed by: 5 members; Thanked: 20 times; GMAT Score:650

by bpolley00 » Wed Jan 23, 2013 5:39 pm

Can someone in more detail touch on being able to put those two together? I am having a tough time conceptually understanding. Also with it not being A, I am assuming we are assuming the photo is not drawn to scale? Is that correct? Thanks!
-BP

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 amFollowed by: 1535 members; Thanked: 2256 times; GMAT Score:800

by lunarpower » Thu Jan 24, 2013 3:17 am

bpolley00 wrote:Can someone in more detail touch on being able to put those two together?

most likely, the notation is at least partly to blame; a lot of people forget that you can type "âˆš" by pressing alt+v (if you have a mac) or, failing that, by just googling "square root sign" and then copying and pasting.

the first statement says that (AC)^2 = 2(AD)^2.
if you take the square root of both sides, you get AC = (âˆš2)AD.

the second statement implies that AC = BC, since those are the only two sides of triangle ABC that can actually be the same. (the hypotenuse of a right triangle can't equal one of the legs.) so, each of AC and BC is equal to (âˆš2)AD.

the only side you're missing now (as far as the sides necessary to find the triangles' areas) is AB.

three ways to go from here:

1/
remember that it's data sufficiency. because triangle ABC is a 45Âº-45Âº-90Âº triangle (isosceles right triangle), you know that it has to have some constant proportions (even if you don't remember what they are). so, AB is some constant multiple of AC or BC, so it's AD times something.
if that's the case, then the area of triangle ABD is (base)(height)/2 = (some # times AD)(AD)/2 = some number times (AD)^2.
the area of triangle ABC is (base)(height)/2 = (ADâˆš2)(ADâˆš2)/2 = AD^2. that's either definitely equal or definitely not equal to the area of ABD (it can't be both), so, sufficient.

2/
using the pythagorean theorem, actually find the value of AB:
(AB)^2 = (ADâˆš2)^2 + (ADâˆš2)^2
this gives AB = 2AD.
so, then, the area of triangle ABD is (base)(height)/2 = (2AD)(AD)/2 = (AD)^2.
this is the same as the area of triangle ABC (calculated above), so you get "yes" = sufficient.

3/
if you have the proportions for 45Âº-45Âº-90Âº triangles memorized, you can also find AB that way.
the hypotenuse of such a triangle is âˆš2 times either of the legs, so AB = (âˆš2)(ADâˆš2) = 2AD.
same as #2 from here.

Ron has been teaching various standardized tests for 20 years.

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 amFollowed by: 1535 members; Thanked: 2256 times; GMAT Score:800

by lunarpower » Thu Jan 24, 2013 3:21 am

bpolley00 wrote: Also with it not being A, I am assuming we are assuming the photo is not drawn to scale? Is that correct? Thanks!
-BP

that is correct. as it seems you've already noticed, once you have both statements, it turns out that AD is much shorter than any other side in the picture -- definitely not the way it looks.

this shouldn't surprise you, though. in fact, NO data sufficiency diagram will ever be "to scale" at the start of the problem!
that may sound like a strange claim, but think about it for a second -- at the beginning of a data sufficiency problem, there are always unknowns, or quantities that can vary, in the diagram. (for instance, in this diagram, there are various pairs of sides that can exist in any imaginable proportion until you restrict them with one or the other of the numbered statements.)
clearly there is no "accurate" depiction of something with many possible appearances, so all data-sufficiency diagrams are "not to scale" by default.

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bpolley00: Master | Next Rank: 500 Posts; Posts: 192; Joined: Wed Jul 11, 2012 5:04 pmFollowed by: 5 members; Thanked: 20 times; GMAT Score:650

by bpolley00 » Thu Jan 24, 2013 5:47 am

Ron,

Thanks so much for the thorough explanation. Your comprehensive understanding of this test and your willingness to openly share that understanding has been instrumental to my comprehension of ways to approach problems I would have never thought of on my own. It is really appreciated.

Thanks
-Bp

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 amFollowed by: 1535 members; Thanked: 2256 times; GMAT Score:800

by lunarpower » Sat Jan 26, 2013 8:50 am

bpolley00 wrote:Ron,

Thanks so much for the thorough explanation. Your comprehensive understanding of this test and your willingness to openly share that understanding has been instrumental to my comprehension of ways to approach problems I would have never thought of on my own. It is really appreciated.

Thanks
-Bp

you're welcome, thanks for reading.

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mcdesty: Master | Next Rank: 500 Posts; Posts: 116; Joined: Wed Mar 14, 2012 1:02 pmFollowed by: 11 members; Thanked: 20 times; GMAT Score:760

by mcdesty » Wed Jul 09, 2014 10:29 am

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