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In quadrilateral ABCD above, what is the length of AB ?

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In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 26 Apr 2019, 01:38
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A
B
C
D
E

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Question Stats:

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In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post Updated on: 05 Aug 2019, 15:31
1
1
Join D and B
DCB is an right angle triangle

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

\(DB^2=9 +16=25\)

\(or DB=5\)

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

\(25=AB^2 + 1\)

\(AB^2 = 24\)

\(AB= 2\sqrt{6}\)

Originally posted by nick1816 on 26 Apr 2019, 02:11.
Last edited by nick1816 on 05 Aug 2019, 15:31, edited 2 times in total.
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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 27 Apr 2019, 05:28
Bunuel wrote:
Image
In quadrilateral ABCD above, what is the length of AB ?


A. \(\sqrt{26}\)

B. \(2\sqrt{5}\)

C. \(2\sqrt{6}\)

D. \(3\sqrt{2}\)

E. \(3\sqrt{3}\)


PS58502.01
OG2020 NEW QUESTION

Attachment:
2019-04-26_1235.png


join BD we get two ∆ right angled BCD and ABD ; BCD ; 3:4:5 and BD = 5 so 25-1 = AB^2
AB - \(2\sqrt{6}\)
IMO C
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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 27 Apr 2019, 17:03
Hi All,

We're asked for the length of AB in quadrilateral ABCD.

When dealing with 'weird' shapes, it often helps to break the shape down into 'pieces' that are easier to deal with. Here, if you draw a line from B to D, you will from 2 RIGHT TRIANGLES.

Triangle BCD has legs of 3 and 4, so it's a 3/4/5 right triangle.
Triangle BAD then has a leg of 1 and a hypotenuse of 5. We can use the Pythagorean Formula to find the missing leg...

1^2 + B^2 = 5^2
1 + B^2 = 25
B^2 = 24

From here, if we square-root both sides, we'll have...
B = √24
B = 2√6

Final Answer:

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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 29 Apr 2019, 10:49
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Top Contributor
Bunuel wrote:
Image
In quadrilateral ABCD above, what is the length of AB ?


A. \(\sqrt{26}\)

B. \(2\sqrt{5}\)

C. \(2\sqrt{6}\)

D. \(3\sqrt{2}\)

E. \(3\sqrt{3}\)


PS58502.01
OG2020 NEW QUESTION

Attachment:
2019-04-26_1235.png

First add a line to join B and D:
Image


If we focus on the blue right triangle, we can EITHER recognize that legs of length 3 and 4 are part of the 3-4-5 Pythagorean triplet, OR we can apply the Pythagorean Theorem.
Image
Either way, we'll see that the triangle's hypotenuse (BD) must have length 5

Now, when we focus on the red right triangle, we can . . .
Image
. . . apply the Pythagorean Theorem to write: x² + 1² = 5²
Simplify: x² + 1 = 25
So: x² = 24
So: x = √24 = √[(4)(6)] = (√4)(√6) = 2√6

Answer: C

Cheers,
Brent

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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 30 Apr 2019, 23:50

Solution



Given:

    • AD= 1
    • CD=1
    • BC=1

To Find:
    • The length of AB.

Approach and Working:

Let us join the points B and D.

Image

Since ∆ BCD is a right -angled triangle, we can apply Pythagoras theorem in ∆ BCD.
    • Thus, \(BD^2\) = \(CD^2\) +\(BC^2\)
      o \(BD^2\) = \(4^2\) +\(3^2\) = 16+9 = 25
      o \(BD^2\) = 25
         BD=5

Now, we can apply Pythagoras theorem in triangle ABD.
    • Hence, \(BD^2\) =\(AD^2\) + \(AB^2\)
      o 25 = 1 + \(AB^2\)
         \(AB^2\) = 24
         AB = √24 = 2√6.

Hence, option C is the correct answer.

Correct Answer: Option C
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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 02 May 2019, 17:27
Bunuel wrote:
Image
In quadrilateral ABCD above, what is the length of AB ?


A. \(\sqrt{26}\)

B. \(2\sqrt{5}\)

C. \(2\sqrt{6}\)

D. \(3\sqrt{2}\)

E. \(3\sqrt{3}\)


PS58502.01
OG2020 NEW QUESTION

Attachment:
2019-04-26_1235.png


If we draw diagonal BD, we’ve created two right triangles: BCD and BAD. We see that triangle BCD is a 3-4-5 right triangle. So we see that side BD = 5.

Therefore, triangle BAD is a right triangle with a leg of 1 and a hypotenuse of 5. We can let side AB = n and use the Pythagorean theorem to determine n.

1^2 + n^2 = 5^2

1 + n^2 = 25

n^2 = 24

n = √24

n = √4 x √6 = 2√6

Answer: C
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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 29 May 2019, 13:23
Bunuel wrote:
Image
In quadrilateral ABCD above, what is the length of AB ?


A. \(\sqrt{26}\)

B. \(2\sqrt{5}\)

C. \(2\sqrt{6}\)

D. \(3\sqrt{2}\)

E. \(3\sqrt{3}\)


PS58502.01
OG2020 NEW QUESTION

Attachment:
2019-04-26_1235.png


If we draw diagonal BD, There are two right triangles: BCD and BAD. We see that triangle BCD is a 3-4-5 right triangle. So we see that side BD = 5.

Therefore, triangle BAD is a right triangle with a leg of 1 and a hypotenuse of 5. Now use the Pythagorean theorem to determine AB.

1^2 + AB^2 = 5^2

1 + AB^2 = 25

AB^2 = 24

AB = √24

AB = √4 x √6 = 2√6

Answer: C

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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 05 Aug 2019, 08:28
HI,
can we also use the 30 : 60 : 90 triangle property to solve for the required side here ?

I tried using, but got a weird answer..
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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 05 Aug 2019, 15:25
Hi Shrey9,

Neither of these triangles is a 30/60/90 right triangle - so that property does not apply. To use 30/60/90 rules, you need to either have the 3 angles or know that you have a right triangle and 2 of the sides that fit the relationship of 2 of the 3 sides of that type of triangle (re. X : X√3 : 2X)

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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 05 Aug 2019, 15:45
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EMPOWERgmatRichC wrote:
Hi Shrey9,

Neither of these triangles is a 30/60/90 right triangle - so that property does not apply. To use 30/60/90 rules, you need to either have the 3 angles or know that you have a right triangle and 2 of the sides that fit the relationship of 2 of the 3 sides of that type of triangle (re. X : X√3 : 2X)

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so what you're saying is Pythagoras can be applied to any right triangle (even if its just one angle, which is 90deg) and 30 :60 : 90 or 45 : 45 : 90 can't just be applied when we don't know other angles apart from the 90 deg one ? (coz it could be 90 : 46 : 44 etc..)
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Re: In quadrilateral ABCD above, what is the length of AB ?  [#permalink]

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New post 05 Aug 2019, 19:48
Hi Shrey9,

That is exactly correct! The Pythagorean Theorem applies to ANY Right Triangle; but the special Right Triangles only occur under specific conditions.

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Re: In quadrilateral ABCD above, what is the length of AB ?   [#permalink] 05 Aug 2019, 19:48
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