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What is CD in the figure above?

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What is CD in the figure above?  [#permalink]

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New post 02 Sep 2016, 05:42
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A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

71% (01:16) correct 29% (01:14) wrong based on 235 sessions

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What is CD in the figure above?  [#permalink]

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New post 02 Sep 2016, 06:11
Bunuel wrote:
Image
What is CD in the figure above?

A. √2
B. 2
C. √6
D. 2√2
E. 4

Attachment:
T6009.png


\((AB)^2\) = \((AD)^2\) + \((BD)^2\)

\((\sqrt{3})^2\) = \((1)^2\) + \((BD)^2\)

\(3\) = \(1\) + \((BD)^2\)

\((BD)^2\) = \(2\)

\((BD)\) = \(\sqrt{2}\)

Now in Δ BCD ; BC = BD and ∠ DBC = 90°

Here , \((DB)^2\) + \((BC)^2\) = \((DC)^2\)

Or, \((\sqrt{2})^2\) + \((\sqrt{2})^2\) = \((DC)^2\)

Or, \(2\) + \(2\) = \((DC)^2\)

Or, \(4\) = \((DC)^2\)

So, \((DC)\) = \(2\)

Hence CD = 2

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Re: What is CD in the figure above?  [#permalink]

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New post 06 Sep 2016, 14:02
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Bunuel wrote:
Image
What is CD in the figure above?

A. √2
B. 2
C. √6
D. 2√2
E. 4

Attachment:
T6009.png


Let x = length of side DB
Image

Since the highlighted triangle is a RIGHT TRIANGLE, we can apply the Pythagorean Theorem.
We get: 1² + x² = (√3)²
Evaluate to get: 1 + x² = 3
So, x² = 2, which means x = √2.....
Image

Now focus on the lower triangle (in blue)
Image

Since we're told that DB = BC, we can see that side BC = √2
We'll let the 3rd side have length y
Image
Since the highlighted triangle is a RIGHT TRIANGLE, we can apply the Pythagorean Theorem
We get: (√2)² + (√2)² = y²
Evaluate to get: 2 + 2 = y²
So, 4 = y², which mean y = 2

Answer:

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Re: What is CD in the figure above?  [#permalink]

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New post 15 Nov 2016, 00:18
fell on trap of 1:sqrt3:2 ratio in which 2 is a hypotenuse, not the leg as considered.

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What is CD in the figure above?  [#permalink]

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New post 05 Jul 2017, 12:29
Once we derive the length of BD as \(\sqrt{2}\) , we can use the property of a right isosceles triangle " Length of the hypotenuse is \(\sqrt{2}a\)"

Hence CD will be\(\sqrt{2} * \sqrt{2}\) = 2

Answer is B
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What is CD in the figure above?  [#permalink]

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New post 05 Jul 2017, 12:46
We need to know the following properties of a right angled triangle.

If the right triangle is isosceles in nature, angle ratio(45:45:90)
the sides are in the ratio 1:1:\(\sqrt{2}\)


Coming back to the question,
Using Pythagoras theorem, \(AB^2 = AD^2 + BD^2\)
From this we can find out that the value of BD = \(\sqrt{2}\)(because BD = \(\sqrt{3-1}\))

In second triangle, using Pythagoras theorem, \(CD^2 = BC^2 + BD^2\)
Since BD = \(\sqrt{2}\) and the right angled triangle is isosceles,
the hypotenuse(CD) = 2(Option B)
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Re: What is CD in the figure above?  [#permalink]

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New post 05 Jul 2017, 12:54
Bunuel wrote:
Image
What is CD in the figure above?

A. √2
B. 2
C. √6
D. 2√2
E. 4

Attachment:
T6009.png


Given Both the triangles are Right Angles Triangles

So, we can write

\((AB)^2 = (AD)^2 + (DB)^2\)

\({\sqrt{3}}^2 = (1)^2 + (DB)^2\)

\((DB)^2 = 3 - 1 = 2\)

\(DB = \sqrt{2}\)

Now for the other triangle

\((CD)^2 = (DB)^2 + (BC)^2\)

As, \(DB = BC\)

\((CD)^2 = {\sqrt{2}}^2 + {\sqrt{2}}^2\)

\((CD)^2 = 2 + 2\)

\((CD)^2 = 4\)

\(CD= 2\)

Hence, Answer is B
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Re: What is CD in the figure above?  [#permalink]

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New post 05 Jul 2017, 16:16
OK to be clear, you cannot use 1: root 3: 2 , because the angles aren't given?
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Re: What is CD in the figure above?  [#permalink]

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New post 05 Jul 2017, 20:10
gmathopeful19 wrote:
OK to be clear, you cannot use 1: root 3: 2 , because the angles aren't given?



Hi..

The reason is not that angles are not given.
BUT the main reason is that in 1:√3:2, the ratio 2 is the hypotenuse.
Here √3 is given as hypotenuse so ratio here is 1:__:√3, which is not same as 1:√3:__
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Re: What is CD in the figure above?  [#permalink]

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New post 06 Jul 2017, 07:54
chetan2u wrote:
gmathopeful19 wrote:
OK to be clear, you cannot use 1: root 3: 2 , because the angles aren't given?



Hi..

The reason is not that angles are not given.
BUT the main reason is that in 1:√3:2, the ratio 2 is the hypotenuse.
Here √3 is given as hypotenuse so ratio here is 1:__:√3, which is not same as 1:√3:__



Would you not just do 2x = root 3 ?

I always took it as 1x : root 3 * x : 2x

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Re: What is CD in the figure above?   [#permalink] 06 Jul 2017, 07:54
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