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

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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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]
Expert Reply

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.



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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Bunuel wrote:

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]
Bunuel wrote:

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]
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]
Expert Reply
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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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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Rich




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]
Expert Reply
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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In quadrilateral ABCD above, what is the length of AB ? [#permalink]
Top Contributor
Bunuel wrote:

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


\(According \ to \ Pythagorean \ triples \ BD=5. \ (A 3:4:5) \ right \ triangle.\)

\((AB)^2+(1)^2=5^2\)

\((AB)^2=24\)

\(AB=\sqrt{24}\)

\(AB=\sqrt{6*4}\)

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

The answer is \(C\)
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Re: In quadrilateral ABCD above, what is the length of AB ? [#permalink]
Expert Reply
Bunuel wrote:

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



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Answer: Option C

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Re: In quadrilateral ABCD above, what is the length of AB ? [#permalink]
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Video solution from Quant Reasoning:
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Re: In quadrilateral ABCD above, what is the length of AB ? [#permalink]
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Re: In quadrilateral ABCD above, what is the length of AB ? [#permalink]
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