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Math Expert V
Joined: 02 Sep 2009
Posts: 62637
What is CD in the figure above?  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 72% (01:14) correct 28% (01:12) wrong based on 196 sessions

### HideShow timer Statistics What is CD in the figure above?

A. √2
B. 2
C. √6
D. 2√2
E. 4

Attachment: T6009.png [ 4.51 KiB | Viewed 3407 times ]

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

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Bunuel wrote: 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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Bunuel wrote: 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 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..... Now focus on the lower triangle (in blue) Since we're told that DB = BC, we can see that side BC = √2
We'll let the 3rd side have length y 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

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

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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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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

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

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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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Bunuel wrote: 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, $$DB = BC$$

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

$$(CD)^2 = 2 + 2$$

$$(CD)^2 = 4$$

$$CD= 2$$

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

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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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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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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]

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_________________ Re: What is CD in the figure above?   [#permalink] 06 May 2019, 10:06
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