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805+ (Hard)|   Geometry|      
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TheUltimateWinner
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In the diagram to the below, the value of x is closest to which of the 11 Mar 2020, 09:18
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37% (02:24) correct 63% (02:37) wrong based on 222 sessions

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In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)
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C

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In the diagram to the below, the value of x is closest to which of the Updated on: 06 May 2021, 02:24
Originally posted by sujoykrdatta on 11 Mar 2020, 12:39.
Last edited by sujoykrdatta on 06 May 2021, 02:24, edited 2 times in total.
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In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)
Show more



Let us drawn AP perpendicular to BC as shown below:

Attachment:
Screenshot_20210506-145136.jpg
Screenshot_20210506-145136.jpg [ 38.57 KiB | Viewed 5110 times ]

Attachment:
111.JPG
111.JPG [ 12.39 KiB | Viewed 5258 times ]

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = \(\sqrt{3}\)

Answer C
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In the diagram to the below, the value of x is closest to which of the Updated on: 11 Mar 2020, 19:36
Originally posted by TheUltimateWinner on 11 Mar 2020, 13:16.
Last edited by TheUltimateWinner on 11 Mar 2020, 19:36, edited 1 time in total.
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sujoykrdatta
Asad
In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)



Let us drawn AP perpendicular to BC as shown below:

Attachment:
111.JPG

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = 2

Answer B
Show more

\(√3.33\) = \(1.8\)
1.8 is closed to \(√3\). So the correct choice is C, actually.
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In the diagram to the below, the value of x is closest to which of the 11 Mar 2020, 13:53
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sujoykrdatta
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In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)



Let us drawn AP perpendicular to BC as shown below:

Attachment:
111.JPG

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = 2

Answer B

\(√3.33\) = \(1.43\)
1.43 is closed to \(√3\). So the correct choice is C, actually.
Show more


How is \(√3.33\) = \(1.43\)?

\(1.4^2 = 1.96\)
\(1.5^2 = 2.25\)
\(1.8^2 = 3.24\)
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In the diagram to the below, the value of x is closest to which of the 11 Mar 2020, 15:12
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IMO ans is 1

Its an iso . triangle bcoz it has 2 equal sides.
One of the angle is 45 so the other one is also 45( same sides)

Its a 45 45 90 triangle.
The sides of 45 45 90 triangles are in ratio of 1 1 root 2.
Hypo is root 2 therefore x is 1

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In the diagram to the below, the value of x is closest to which of the 11 Mar 2020, 16:55
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sujoykrdatta
Asad
In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)



Let us drawn AP perpendicular to BC as shown below:

Attachment:
111.JPG

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = 2

Answer B
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Hi sir, I think its closer to \(\sqrt{3}\).
\(\sqrt{3}\)=1.7. Your answer is 1.8. It is closer to 1.7 than 2.

Anyways my working is as follows -
\(2 = (\frac{x}{\sqrt{2}})^2 + (x - \frac{x}{\sqrt{2}})^2\)
\(or, 2 = 2x^2 -\frac{2x^2}{\sqrt{2}}\)
\(or, 1 = x^2 - \frac{x^2}{\sqrt{2}}\)
\(or, x^2 = \frac{\sqrt{2}}{(\sqrt{2}-1)}\)
\(or, x^2 = 1.4/0.4 = 3 (approx)\)
\(or, x = \sqrt{3}\)
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In the diagram to the below, the value of x is closest to which of the 11 Mar 2020, 18:47
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AnirudhaS
sujoykrdatta
Asad
In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)



Let us drawn AP perpendicular to BC as shown below:

Attachment:
111.JPG

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = 2

Answer B
Hi sir, I think its closer to \(\sqrt{3}\).
\(\sqrt{3}\)=1.7. Your answer is 1.8. It is closer to 1.7 than 2.

Anyways my working is as follows -
\(2 = (\frac{x}{\sqrt{2}})^2 + (x - \frac{x}{\sqrt{2}})^2\)
\(or, 2 = 2x^2 -\frac{2x^2}{\sqrt{2}}\)
\(or, 1 = x^2 - \frac{x^2}{\sqrt{2}}\)
\(or, x^2 = \frac{\sqrt{2}}{(\sqrt{2}-1)}\)
\(or, x^2 = 1.4/0.4 = 3 (approx)\)
\(or, x = \sqrt{3}\)
Show more


Yes obviously :roll:
I don't know what came of me :|
Must have not seen that option of root3 at all! :-D - corrected that - thanks

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In the diagram to the below, the value of x is closest to which of the 11 Mar 2020, 19:44
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sujoykrdatta



Let us drawn AP perpendicular to BC as shown below:

Attachment:
111.JPG

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = 2

Answer B

\(√3.33\) = \(1.43\)
1.43 is closed to \(√3\). So the correct choice is C, actually.


How is \(√3.33\) = \(1.43\)?

\(1.4^2 = 1.96\)
\(1.5^2 = 2.25\)
\(1.8^2 = 3.24\)
Show more
My calculator gave me wrong info (value of root 3.33). I was convinced with that wrong value (1.43) because i already know that the correct choice is C.

Edited the first comment..

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In the diagram to the below, the value of x is closest to which of the 29 Mar 2020, 07:15
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I do not understand this question. When x and x are the same lengths, their respective angles have to have the same degree, right ? so i do not understand how anyonecan say this is a 45 - 45 - 90 triangle. If this was true root 2 has to be the longest side, so that the angles of 45 and 45 are respective to the two x.

In this case side root 2 faces the angle 45 so the angles for x are 135/2, so you can not use the pythagorean theorem to calculate the sides?
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In the diagram to the below, the value of x is closest to which of the 29 Apr 2021, 09:01
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I believe that the question was created by Manhattan Prep to illustrate strategic guessing (I could be wrong, it was a while ago). Also, since we are estimating, we can make some approximate guesses.

I’m a triangle, the sides of a triangle are, in a sense, proportional to the angles of the triangle. The sides are in fact in a constant proportion to the Sine of each angle’s degree measure - the angle opposite each Side

Side a / (Sin Angle Opp a) = Side b / (Sin Angle Opp b) = Side c / (Sin Angle Opp c)

(1st) given we have an isosceles triangle, the other 2 angle measures both equal = (135/2) = 67.5 degrees


(2nd) The “Sine” Proportion can be taken. For each side and its opposite angle, the following proportion holds true:

Sqrt(3) / (Sine 45 deg.) = (X) / (Sine 67.5 deg.) = (X) / (Sine 67.5 deg.)


The Sine of 45 degrees = (1) / (sqrt(2))

We can use the Sine of 60 degrees to approximate the Sine of 67.5 degrees, since it easier to find and since 60 degrees is close to 67.5 degrees

The sine of 60 degrees = sqrt(3) / 2


Substituting we have:

[ sqrt(2) ] / [ 1/sqrt(2) ] = [ x ] / [ sqrt(3) / 2]

In the right hand side of the equation, the square root of 2 squared = 2——-you are left with:


2 = (2x) / (sqrt(3))

—cancel the 2 in the NUM on each side of the equation—

1 = X / (sqrt(3))

-cross multiply-

X = sqrt(3)

Since this is only an approximation and not the actual value or X, I suspect there are other ways to estimate the answer. It’s too coincidental that the estimate found happened to be the exact value of the correct answer approximation.

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In the diagram to the below, the value of x is closest to which of the 06 May 2021, 02:23
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In the diagram to the below, the value of \(x\) is closest to which of the following?

A) \(2+√2\)
B) \(2\)
C) \(√3\)
D) \(√2\)
E) \(1\)



Let us drawn AP perpendicular to BC as shown below:

Attachment:
The attachment 111.JPG is no longer available

In triangle APC: Angle ACP = Angle CAP = \(45^o\)

=> It is a 45-45-90 triangle

=> AP = PC = AC/\(\sqrt{2}\) = \(x/\sqrt{2}\)

=> BP = \(x - x/\sqrt{2}\)

In right triangle ABP: \(AB^2 = AP^2 + BP^2\)

=> \(2 = (x/\sqrt{2})^2 + (x - x/\sqrt{2})^2\)

=> \(2 = x^2/2 + x^2 + x^2/2 - 2x^2/\sqrt{2}\)

=> \(2 = x^2 * (2 - \sqrt{2})\)

=> \(x^2 = 2/(2 - \sqrt{2}) = ~ 2/0.6 = 3.33\) (approximately)

=> \(x = \sqrt{3.33} = 1.8\)

=> Closest value = \(\sqrt{3}\)

Answer C
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For some reason the image is not visible ... Uploading here once more:

Attachment:
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Screenshot_20210506-145136.jpg [ 38.57 KiB | Viewed 4757 times ]

Hope this helps.
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In the diagram to the below, the value of x is closest to which of the 25 Jul 2021, 21:53
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I had no idea how to solve this in a methodological way so I took an educated guess. In a 45-45-90 triangle, the largest side is √2 times the smallest side. The smallest side in our question is √2, so we know that the largest side (2 in our case) cannot be 2 (√2*√2) since this is not a 45-45-90 triangle. The largest side will have to be less than 2 and obviously more than √2 (the side opposite the smaller angle). We're only left with option C here. Is this approach correct?
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