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805+ Level|   Geometry|      
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annmary
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In the diagram to the right, what is the value of x? Updated on: 10 May 2022, 01:58
Originally posted by annmary on 06 Jun 2011, 03:18.
Last edited by Bunuel on 10 May 2022, 01:58, edited 4 times in total.
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B
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In the diagram to the right, what is the value of x?


A. \(1+\sqrt 2\)

B. \(1+\sqrt 3\)

C. \(2\sqrt 2\)

D. \(\sqrt 2 + \sqrt 3\)

E. \(2\sqrt 3\)

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In the diagram to the right, what is the value of x? 31 Mar 2012, 14:16
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In the diagram to the right, what is the value of x?

A. \(1+\sqrt 2\)

B. \(1+\sqrt 3\)

C. \(2\sqrt 2\)

D. \(\sqrt 2 + \sqrt 3\)

E. \(2\sqrt 3\)

Check below:



Triangle BDC is a 30°-60°-90° right triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\), (the leg opposite 30° (BD) corresponds with \(1\), the leg opposite 60° (DC) corresponds with \(\sqrt{3}\), and the hypotenuse \(BC=x\) corresponds with 2). Thus, since \(BC=x\) then \(BD=\frac{x}{2}\) and \(DC=\frac{x\sqrt{3}}{2}\).

Also notice that \(AD=AC-DC=x-\frac{x\sqrt{3}}{2}\).

Next, from Pythagoras theorem \(AB^2=BD^2+AD^2\);

\(2=(\frac{x}{2})^2+(x-\frac{x\sqrt{3}}{2})^2\);

\(2=x^2(\frac{1}{4}+1-2*\frac{\sqrt{3}}{2}+\frac{3}{4})\);

\(2=x^2(2-\sqrt{3})\);

\(x^2=\frac{2}{2-\sqrt{3}}\);

Rationalize: \(x^2=\frac{2*(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=4+2\sqrt{3}\);

\(x^2=4+2\sqrt{3}=(\sqrt{3})^2+2\sqrt{3}+1=(\sqrt{3}+1)^2\);

\(x=\sqrt{3}+1\).

Answer: B.

Hope it's clear.

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In the diagram to the right, what is the value of x? 06 Jun 2011, 19:28
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In the diagram to the right, what is the value of x?

A. \(1+\sqrt 2\)

B. \(1+\sqrt 3\)

C. \(2\sqrt 2\)

D. \(\sqrt 2 + \sqrt 3\)

E. \(2\sqrt 3\)


They have given a triangle in the question and a 30 degree angle but no right triangle. So, my first instinct is to drop an altitude (a) to get a 30-60-90 triangle.

Attachment:
Ques2.jpg
Ques2.jpg [ 7.13 KiB | Viewed 14165 times ]

We know that the sides of a 30-60-90 triangle are \(a\), \(\sqrt{3}a\) and \(2a\).
Hence the smaller right triangle has sides \(a\), \(2a - \sqrt{3}a\) and \(\sqrt{2}\).

Using Pythagorean theorem,

\(a^2 + (2a - \sqrt{3}a)^2 = \sqrt{2}^2\).

\(a^2 = \frac{1}{4 - 2\sqrt{3}} = \frac{1}{(\sqrt{3} - 1)^2}\)

\(a = x/2 = \frac{1}{(\sqrt{3} - 1)}\)

\(x = \sqrt{3} + 1\)
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In the diagram to the right, what is the value of x? 03 Aug 2011, 18:09
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How do you go from
\(a^2 + (2a - \sqrt{3}a)^2 = \sqrt{2}^2\).

to
\(a^2 = \frac{1}{4 - 2\sqrt{3}} = \frac{1}{(\sqrt{3} - 1)^2}\)
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In the diagram to the right, what is the value of x? 03 Aug 2011, 20:59
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Yalephd
How do you go from
\(a^2 + (2a - \sqrt{3}a)^2 = \sqrt{2}^2\).

to
\(a^2 = \frac{1}{4 - 2\sqrt{3}} = \frac{1}{(\sqrt{3} - 1)^2}\)
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\(a^2 + (2a - \sqrt{3}a)^2 = \sqrt{2}^2\)
\(a^2 + a^2(2 - \sqrt{3})^2 = 2\)
\(a^2(1 + (2 - \sqrt{3})^2) = 2\)
\(a^2(1 + 4 + 3 - 4\sqrt{3}) = 2\) (Using \((a + b)^2 = a^2 + b^2 + 2ab\))
\(a^2 = 2/(8 - 4\sqrt{3})\)
\(a^2 = 1/(4 - 2\sqrt{3})\)

Now, \((4 - 2\sqrt{3})\) = \((1 + 3 - 2\sqrt{3})\) = \((1^2 + \sqrt{3}^2 - 2*1*\sqrt{3})\)
= \((\sqrt{3} - 1)^2\)

Therefore, \(a^2 = 1/(\sqrt{3} - 1)^2\)
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In the diagram to the right, what is the value of x? 09 Aug 2011, 21:20
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We can also use the law of cosines to arrive at the solution quickly:
a^2 = b^2 + c^2 - 2bc(cos A) [here a,b,and c are the sides of the triangle and A is the angle opposite a]

Applying it to this question:
=> {sqrt(2)}^2 = x^2 + x^2 - 2x^2 (cos 30)
=> 2 = 2x^2 - 2x^2 (sqrt(3)/2)
=> 2 = 2x^2 - sqrt(3)*x^2
=> x^2 = 2/(2 -sqrt(3))= 2(2 + sqrt(3))/(4-3)
=> x^2 = 4 + sqrt(3) = 1 + 3 + 2*sqrt(3) = (1+sqrt(3))^2
=> x = 1+ sqrt(3)
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In the diagram to the right, what is the value of x? 17 Apr 2019, 09:02
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The MGMAT book also talks about estimating in a problem like this. Assume the diagram is drawn with the altitude added as above to make a 30:60:90 triangle:

√2 ≈ 1.4 (a bit more than) Now, obviously the perpendicular line is a bit less than this. However, if we assume that its length is also 1.4 then we can use the 30:60:90 triangle ratios right away and eliminate some answers.

If 1.4 = s (angle with 30) then x = 2s, or 2.8 So we know our maximum possible value is a bit less than this. Right away we can eliminate C, D and E.

C) 2*1.4 = 2.8
D) 1.4 + 1.7 = 3.1
E) 2*1.7 = 3.4

We're left with A and B.
A) 1+1.4 = 2.4
B) 1+1.7 = 2.7

Here I would just guess and move on, we assumed the line is just a bit less than 1.4 so B) is closest. You can also note that the answers vary by .3 except for B and C which can potentially be a hint (or not...)
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In the diagram to the right, what is the value of x? 09 Aug 2023, 12:48
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