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02 Dec 2019, 00:34
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
31% (02:20) correct
69%
(01:49)
wrong
based on 75
sessions
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Not Attempted Yet
In the above triangle, if \(AB = 6\), what is the length of AD ?
(1) \(BD=2\sqrt{5}\)
(2) \(DC=5\)
Attachment:
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Updated on: 05 Dec 2019, 03:59
Originally posted by rajatchopra1994 on 02 Dec 2019, 03:44.
Last edited by rajatchopra1994 on 05 Dec 2019, 03:59, edited 1 time in total.
Last edited by rajatchopra1994 on 05 Dec 2019, 03:59, edited 1 time in total.
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Bunuel
Explanation:
Given AB=6, AD=?
Statement 1:
In Triangle ABD, Angle BDC is 90 Degree, also Given \(BD=2\sqrt{5}\)
We can find 3rd side of right angle Triangle ABD
Sufficient
Statement 2: DC=5
IMO
Please give kudos, if you find my explanation good enough 
The answer is B D.
We do have two properties of a right triangle when we have an altitude drawn from the right angle to hypotenuse:
1) The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.
In terms of the right triangle in question: BD=\sqrt{AD*DC}
2) If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
In terms of the right triangle in question: AB =\sqrt{AD*AC}
So, to find AD, we need either AC or BD and DC
From the second property, AB^2 = AD*DC >> 6^2 = AD*(AD+5) >> AD^2+5AD-36=0 >> AD = 4 or AD=-5. We can accept only positive numbers, so AD=4.
We do have two properties of a right triangle when we have an altitude drawn from the right angle to hypotenuse:
1) The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.
In terms of the right triangle in question: BD=\sqrt{AD*DC}
2) If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
In terms of the right triangle in question: AB =\sqrt{AD*AC}
So, to find AD, we need either AC or BD and DC
Quote:
Sufficient. We can simple apply Pythagoras theorem (thanks again, @rajatchopra1994!)
Quote:
Sufficient.
From the second property, AB^2 = AD*DC >> 6^2 = AD*(AD+5) >> AD^2+5AD-36=0 >> AD = 4 or AD=-5. We can accept only positive numbers, so AD=4.
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