Advanced GMAT Quant / Chapter-9 / Q-139 / Page 139, 364, 365

[supratim7]

Advanced GMAT Quant / Chapter-9 / Q-139 / Page 264, 364-66
http://www.filedropper.com/img00229-20130301-2029

Given angle DCB = 90º
Given length DQ / length QB = 1/2

Does this imply that angle DCQ = 30º and angle QCB = 60º ?
i.e. angle DCQ / angle QCB = 1/2 ?

Many thanks | Supratim

[RonPurewal]

no, it doesn't. in fact, you can prove that those angles are not 30º and 60º.

here's one way you can do that: draw in an extra vertical line from point Q down to the bottom of the square, and label "X" where that vertical line hits the bottom of the square.

once you have that, XC = √2 and QX = √2/2 (try to prove these yourself, if you haven't already). that's a ratio of 2:1, which is the wrong ratio for the legs in a 30º-60º triangle.

[supratim7]

Appreciate it Ron. Learnt something new :)

Yes, I did try dropping a vertical from Q to X. Got same values, XC = √2 and QX = √2/2, wrong ratio for a 30º-60º-90º triangle.

So, Diagonal DQ / QB = 1/2, Base DX / XC = 1/2, Height CY / YB = 1/2. But the angles involved are not in that ratio.. right??

Must say, the whole thing is bit counter intuitive :) More so because if DQ / QB = 1, i.e. QC is the diagonal bisector of DB, then, I think, angle DCQ = 45º and angle QCB = 45º.. right??

[RonPurewal]

Must say, the whole thing is bit counter intuitive :)

i wouldn't say it's counterintuitive, unless you go in with the default assumption that angles should obey the same mathematical relationships that lengths do. that would be a rather dubious assumption, considering that angles and lengths are entirely different things.

more generally, you shouldn't assume similar mathematical behavior between different quantities, unless you have some good, solid reason for that assumption.
e.g.
if you shoot 2 things into the air, one with twice the initial speed of the other, does the first one go twice as far upward? nope.
if you slam on the brakes at 80 miles an hour, will you travel twice as far before stopping as you would if you slammed on the brakes at 40 miles per hour? definitely not (and you'll get into a lot of rear-end accidents if you think so).
etc.

[supratim7]

Thank you Ron. Cool illustrations :)

Many thanks | Supratim

[tim]

:)