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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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One side length of a left square is m and one side length of a right s  [#permalink]

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19 00:00

Difficulty:   65% (hard)

Question Stats: 62% (02:53) correct 38% (02:42) wrong based on 162 sessions

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One side length of a left square is m and one side length of a right square is n. What is the area of a middle square, in terms of m and n?
A. m^2+n^2
B.m^2-n^2
C. m+n^2
D. m^2+n
E. m+n

*An answer will be posted in 2 days.

Attachments 1.png [ 12.98 KiB | Viewed 2683 times ]

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One side length of a left square is m and one side length of a right s  [#permalink]

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What an interesting question!

I would approach it in the following way, I hope someone finds it useful. Please refer to the attached image for reference.

We know that
ΔCDG: CD=BA=m, ∠CDF = 90° & CG = CF
ΔFHG: FH=IK=n, ∠FHG = 90° & CG = CF

Now we know that:
CG = CF = m * DG = n * GH

Since both hypothenuses of those two right triangles are equal, we can conclude that they both must have the same lenghts of two legs of a right triangle, thus
CD = GH = m
FH = DG = n

No we can find the side of a square CEFG, which is CG = FG = $$\sqrt{{m^2 + n^2}}$$
The area of the square will equal to CG * FG =$$m^2 + n^2$$

A
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8017
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: One side length of a left square is m and one side length of a right s  [#permalink]

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For this question, you have to utilize the Pythagorean Theorem twice. Then, the answer becomes A.
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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Did not follow your explanation.You have made some mistakes which I have highlighted and also I couldn't understand as to whyCG=GF=m*DG=n*GH

fantaisie wrote:
What an interesting question!

I would approach it in the following way, I hope someone finds it useful. Please refer to the attached image for reference.

We know that
ΔCDG: CD=BA=m, ∠CDF = 90° & CG = CF
ΔFHG: FH=IK=n, ∠FHG = 90° & CG = CF

Now we know that:
CG = CF = m * DG = n * GH

Since both hypothenuses of those two right triangles are equal, we can conclude that they both must have the same lenghts of two legs of a right triangle, thus
CD = GH = m
FH = DG = n

No we can find the side of a square CEFG, which is CG = FG = $$\sqrt{{m^2 + n^2}}$$
The area of the square will equal to CG * FG =$$m^2 + n^2$$

A
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Posts: 27
Re: One side length of a left square is m and one side length of a right s  [#permalink]

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MathRevolution wrote:
For this question, you have to utilize the Pythagorean Theorem twice. Then, the answer becomes A.

Can you please explain how A is the answer.
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Posts: 3
Re: One side length of a left square is m and one side length of a right s  [#permalink]

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How can CG = CF. CG is side of a Square and CF is diagonal. They can never be equal. Can you please explain.

fantaisie wrote:
What an interesting question!

I would approach it in the following way, I hope someone finds it useful. Please refer to the attached image for reference.

We know that
ΔCDG: CD=BA=m, ∠CDF = 90° & CG = CF
ΔFHG: FH=IK=n, ∠FHG = 90° & CG = CF

Now we know that:
CG = CF = m * DG = n * GH

Since both hypothenuses of those two right triangles are equal, we can conclude that they both must have the same lenghts of two legs of a right triangle, thus
CD = GH = m
FH = DG = n

No we can find the side of a square CEFG, which is CG = FG = $$\sqrt{{m^2 + n^2}}$$
The area of the square will equal to CG * FG =$$m^2 + n^2$$

A
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8017
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: One side length of a left square is m and one side length of a right s  [#permalink]

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atomicmass wrote:
MathRevolution wrote:
For this question, you have to utilize the Pythagorean Theorem twice. Then, the answer becomes A.

Can you please explain how A is the answer.

Thank you for the question! Please refer to the attachment.
Attachments 2.jpg [ 2.02 MiB | Viewed 2352 times ] 1.jpg [ 2.02 MiB | Viewed 2356 times ]

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One side length of a left square is m and one side length of a right s  [#permalink]

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Attachment: FullSizeRender.jpg [ 769.21 KiB | Viewed 2038 times ] There is no way I could have solved this in 2 mins Intern  Joined: 19 Jun 2016
Posts: 2
Re: One side length of a left square is m and one side length of a right s  [#permalink]

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This question can actually become very easy.
The trick on these geometry questions is that you can assume things to your convenience, as long as it does not violate any of the conditions on the question.
We can easily assume m=n, we don't violate any conditions on the qus. The answer is pretty straightforward then.
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One side length of a left square is m and one side length of a right s  [#permalink]

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One side length of a left square is m and one side length of a right square is n. What is the area of a middle square, in terms of m and n?

Please refer the attached diagram.
Given that $$BC = m$$ and $$EH = n$$

To solve this question use the fact that middle figure is also square. => BJ = JH => $$CJ = n$$ and $$JE = m$$.

Hence, $$BJ = \sqrt{m^{2} + n^{2}}$$ and area of middle square = $${m^{2} + n^{2}$$
Attachments Square1_MathRevolution.jpeg [ 6.13 KiB | Viewed 1888 times ]

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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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3
The simplest way to solve this question is to assume the base angle in left right angled triangle as x. Since angle in square is 90deg. So the base angle in the right triangle is 90-x. Now(for left triangle) sin(90-x)=cosx = m/a and (for right triangle) sin x=n/a. Now use cos^2 x + sin^2 x =1. And you have result without Pythagoras theorem.
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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ganand wrote:
One side length of a left square is m and one side length of a right square is n. What is the area of a middle square, in terms of m and n?

Please refer the attached diagram.
Given that $$BC = m$$ and $$EH = n$$

To solve this question use the fact that middle figure is also square. => BJ = JH => $$CJ = n$$ and $$JE = m$$.

Hence, $$BJ = \sqrt{m^{2} + n^{2}}$$ and area of middle square = $${m^{2} + n^{2}$$

Hi,

Can someone please explain how did we go from BJ = CJ to CJ = n and JE = m ?
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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is that because the two triangles are congruent ?
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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I don't understand how CG=CF?
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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Can anybody share a simple solution to this problem
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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Based on the diagram- m^2 < n^2. Hence eliminate option B.
In all the other options the dimensions are not correct- meaning- area has to be of the form (length)^2. None of the other options are of that form (you can't do m + n^2 or n + m^2, it's like adding 1 metre and 1 sq.metre)

Hence answer has to be Option A.

Using this method you can solve this in 20 seconds max.
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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My approach.
Name LHS Triangle as ABC with B as 90 deg and RHS Triangle as EDC with angle D as 90 deg.
In LHS triangle ABC
Angle B= 90 deg, Assume angle ACB=x, then angle BAC=90-x. -------------(1)
Angle ACE = 90 deg which means angle ECD = 90-x (Angles ACB, ACE and ECD add up to 180 deg). -------------(2)
Now RHS triangle EDC,
Angle D = 90 deg, angle ECD = 90-x and angle CED = x -------------------(3)
From (1) & (2), triangles ABC and EDC are similar triangles.
AC/BC = CE/ED
a/BC=a/n, BC = n.
Also, AC^2 = AB^2+BC^2, a^2 = m^2+n^2. And we know that a^2 is the area of the middle square. So Area of the middle square is m^2+n^2.
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Re: One side length of a left square is m and one side length of a right s  [#permalink]

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phoenix128 wrote:
Based on the diagram- m^2 < n^2. Hence eliminate option B.
In all the other options the dimensions are not correct- meaning- area has to be of the form (length)^2. None of the other options are of that form (you can't do m + n^2 or n + m^2, it's like adding 1 metre and 1 sq.metre)

Hence answer has to be Option A.

Using this method you can solve this in 20 seconds max.

Brilliant solution, i tried using Pytha and got lost... then tried to make it easier by assuming M = N and viola got the answer..
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