Geometry Trisected
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In the figure above, ABCD is a square, and the two diagonal lines divide it into three regions of equal area. If AB = 3, what is the length of w, the perpendicular distance between the two diagonal lines?
A) 3√2 - 2√3
B) 3√2 - √6
C) √2
D)
E) 2√3 - √6
Time yourself when solving this, and do post your time ! In exam conditions it took me 3.2 Minutes and then I had to guess and move on !
After reviewing the problem in untimed conditions I found out that there are a few "very easy under 2 minutes" solutions available.
[spoiler]OA A : 3√2 - 2√3[/spoiler]
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mevicks wrote:
In the figure above, ABCD is a square, and the two diagonal lines divide it into three regions of equal area. If AB = 3, what is the length of w, the perpendicular distance between the two diagonal lines?
A) 3√2 - 2√3
B) 3√2 - √6
C) √2
D)
E) 2√3 - √6
The area of square ABCD = 3² = 9.
Since the square is divided into 3 equal regions, the area of triangle DWY = 3.
Thus:
(1/2)(DY)(WD) = 3.
Since triangle DWY is isosceles, DY=WD.
Thus:
(1/2)(WD)(WD) = 3
WD² = 6
WD = √6.
Since AD = 3 and WD = √6, AW = 3 - √6.
Triangle AXW is a 45-45-90 triangle.
(Don't worry about a proof. It should be clear from the figure that AXW is an isosceles right triangle.)
Since the sides in triangle AXW are proportioned 1:1:√2, we get:
WX = (3 - √6)√2 = 3√2 - √12 = 3√2 - 2√3.
The correct answer is A.
Last edited by GMATGuruNY on Fri Sep 20, 2013 5:47 am, edited 1 time in total.
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Nice alternate explanation rahul. Although I have a doubt regarding the step in red. How did you find the value of the perpendicular in one step ? (I assume you have considered the area = 1/2*Base*Height formula to find it).
The explanation by Mitch is the fastest and the easiest though.
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mevicks wrote:Nice alternate explanation rahul. Although I have a doubt regarding the step in red. How did you find the value of the perpendicular in one step ? (I assume you have considered the area = 1/2*Base*Height formula to find it).
The explanation by Mitch is the fastest and the easiest though.
Mevrick, there's one rule; The rule says:
where H is the Perpendicular on Hypotenuse
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Any side of a triangle can be deemed THE BASE.
Each base has a corresponding HEIGHT.
Since the area of the triangle must be the same no matter which base and height are used, bh must be THE SAME in each case.
Triangle ADY:
If DY is the base, then DA is the corresponding height.
If AY is the base, let h = the corresponding height (the perpendicular from D to DY).
Since the product must be the same in each case, we get:
(DY)(DA) = (AY)(h)
(√6)(√6) = (√12)(h)
h = √36/√12 = √3.
Last edited by GMATGuruNY on Fri Sep 20, 2013 6:07 am, edited 1 time in total.
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Is this rule valid for all types of Right angled triangles or just Isosceles Right angled triangle ones ? Seems very handy, especially during crunch time :twisted:
Edit: Oh! got it. (1/2)*b*h = (1/2)*b*h = Area
i.e. bh = bh and thus H= ab/c
No matter what the base and height is. So it should work for all right angled triangles
Thanks to Mitch n Rahul for the explainations!
Last edited by mevicks on Fri Sep 20, 2013 6:15 am, edited 1 time in total.
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Yes, I have solved many questions considering this rule. There is no such constraint on the values of a,b, and c; they may varymevicks wrote:
Is this rule valid for all types of Right angled triangles or just Isosceles Right angled triangle ones ? Seems very handy, especially during crunch time :twisted:
Yeah it's a very Handy rule!!!.
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I feel this is like this:mevicks wrote:
In the figure above, ABCD is a square, and the two diagonal lines divide it into three regions of equal area. If AB = 3, what is the length of w, the perpendicular distance between the two diagonal lines?
A) 3√2 - 2√3
B) 3√2 - √6
C) √2
D)
E) 2√3 - √6
Time yourself when solving this, and do post your time ! In exam conditions it took me 3.2 Minutes and then I had to guess and move on !
After reviewing the problem in untimed conditions I found out that there are a few "very easy under 2 minutes" solutions available.
[spoiler]OA A : 3√2 - 2√3[/spoiler]
The length of the diagonal of a square is d = √2
So d = 3√2
Since it is 45-45-90 triangle on both sides, the third side or hypotenuse has to √2, so 2 triangles, hence 2√2. The length w = 3√2-2√2.
If I am wrong, where am i wrong?
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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Vinay, applied a rule with a wrong assumption:vinay1983 wrote:
I feel this is like this:
The length of the diagonal of a square is d = √2
So d = 3√2
Since it is 45-45-90 triangle on both sides, the third side or hypotenuse has to √2, so 2 triangles, hence 2√2. The length w = 3√2-2√2.
If I am wrong, where am i wrong?
In a 45-90-45 triangle,
two non-hypotenuse sides are a & a and not 1 & 1.
So, the Hypotenuse you calculated has to be √2a and not √2.
You need to calculate the measurement of "a"; refer solutions provided by Mitch and me.
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My nemesis!Got it!theCodeToGMAT wrote:Vinay, applied a rule with a wrong assumption:vinay1983 wrote:
I feel this is like this:
The length of the diagonal of a square is d = √2
So d = 3√2
Since it is 45-45-90 triangle on both sides, the third side or hypotenuse has to √2, so 2 triangles, hence 2√2. The length w = 3√2-2√2.
If I am wrong, where am i wrong?
In a 45-90-45 triangle,
two non-hypotenuse sides are a & a and not 1 & 1.
So, the Hypotenuse you calculated has to be √2a and not √2.
You need to calculate the measurement of "a"; refer solutions provided by Mitch and me.
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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I think I wrote this problem, but I honestly can't remember.
In that picture for this problem, the yellow thing divides XDY into two 45º-45º-90º triangles. Therefore, the yellow thing is just DX or DY divided by √2.
You already know DX = DY = √6, so the yellow thing is √3.
This is true, but you don't need this much ammunition here.
In that picture for this problem, the yellow thing divides XDY into two 45º-45º-90º triangles. Therefore, the yellow thing is just DX or DY divided by √2.
You already know DX = DY = √6, so the yellow thing is √3.
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Also, you can solve this problem by just looking at the picture, estimating the length, and estimating the answer choices.
The picture is accurate, so we can estimate visually. Given that AB = 3, a quick glance makes it clear that w is 1 or less.
Approximately 4.2 - 3.4.
Approximately 0.8.
Pretty good.
Whoa! Way way way too big.
Way too big.
More than 2!
Way way way too big.
Approximately 3.4 - (something about halfway between 1 and 2).
Almost 2!
Way way way too big.
A is the only choice that's even halfway reasonable. If you figure out about how big the other four choices are, they're all ridiculous.
The picture is accurate, so we can estimate visually. Given that AB = 3, a quick glance makes it clear that w is 1 or less.
Approximately 3(1.4) - 2(1.7).mevicks wrote:A) 3√2 - 2√3
Approximately 4.2 - 3.4.
Approximately 0.8.
Pretty good.
Approximately 4.2 - (something about halfway between 1 and 2).B) 3√2 - √6
Whoa! Way way way too big.
About 1.4.C) √2
Way too big.
Approximately one and a half times 1.4.(D) 3√2/2
More than 2!
Way way way too big.
Approximately 2(1.7) - (something about halfway between 1 and 2).E) 2√3 - √6
Approximately 3.4 - (something about halfway between 1 and 2).
Almost 2!
Way way way too big.
A is the only choice that's even halfway reasonable. If you figure out about how big the other four choices are, they're all ridiculous.
Ron has been teaching various standardized tests for 20 years.
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Answer: Option Amevicks wrote:
In the figure above, ABCD is a square, and the two diagonal lines divide it into three regions of equal area. If AB = 3, what is the length of w, the perpendicular distance between the two diagonal lines?
A) 3√2 - 2√3
B) 3√2 - √6
C) √2
D)
E) 2√3 - √6
Time yourself when solving this, and do post your time ! In exam conditions it took me 3.2 Minutes and then I had to guess and move on !
After reviewing the problem in untimed conditions I found out that there are a few "very easy under 2 minutes" solutions available.
[spoiler]OA A : 3√2 - 2√3[/spoiler]
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