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A
B
C
D
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 Brent@GMATPrepNow
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[spoiler=] [/spoiler][/quote]
First add a line to join B and D:
If we focus on the blue right triangle, we can EITHER recognize that legs of length 3 and 4 are part of the 345 Pythagorean triplet, OR we can apply the Pythagorean Theorem.
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 . . .
. . . apply the Pythagorean Theorem to write: xÂ² + 1Â² = 5Â²
Simplify: xÂ² + 1 = 25
So: xÂ² = 24
So: x = âˆš24 = âˆš[(4)(6)] = (âˆš4)(âˆš6) = 2âˆš6
Answer: C
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Brent
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If we draw diagonal \(BD\), we've created two right triangles: \(BCD\) and \(BAD\). We see that triangle \(BCD\) is a \(345\) 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 \(n\).
$$
\begin{align*}
1^2 + n^2 &= 5^2\\
1 + n^2 &= 25\\
n^2 &= 24\\
n &= \sqrt{24}\\
&= \sqrt{4} \cdot \sqrt{6} \\
&= 2\sqrt{6}
\end{align*}
$$
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 \(n\).
$$
\begin{align*}
1^2 + n^2 &= 5^2\\
1 + n^2 &= 25\\
n^2 &= 24\\
n &= \sqrt{24}\\
&= \sqrt{4} \cdot \sqrt{6} \\
&= 2\sqrt{6}
\end{align*}
$$
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 Rich.C@EMPOWERgmat.com
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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 squareroot both sides, we'll have...
B = âˆš24
B = 2âˆš6
Final Answer: C
GMAT assassins aren't born, they're made,
Rich
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 squareroot both sides, we'll have...
B = âˆš24
B = 2âˆš6
Final Answer: C
GMAT assassins aren't born, they're made,
Rich