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04 Feb 2022, 03:20
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Difficulty:
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In the triangle above, does AB = BC?
(1) x > y
(2) d > e
Attachment:
Triangle_A.B.C..png [ 3.39 KiB | Viewed 1506 times ]
kungfury42
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Updated on: 04 Feb 2022, 05:49
Originally posted by kungfury42 on 04 Feb 2022, 04:05.
Last edited by kungfury42 on 04 Feb 2022, 05:49, edited 1 time in total.
Last edited by kungfury42 on 04 Feb 2022, 05:49, edited 1 time in total.
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Let the landing point of the altitude from B to AC be D.
(1) x>y
We know that AB*cos(x) = BC*cos(y) = BD
AB/BC = cos(y)/cos(x) and since y < x, cos(y) > cos(x) or cos(y)/cos(x) > 1 or AB/BC > 1 or AB>BC
Clearly Sufficient. (Infact we didn't have to calculate the exact relation between AB & BC, since we had been given x>y we can infer x≠y and thus cos(x)≠cos(y) and thus AB≠BC)
(2) d > e
We know that AB² = BD² + d² and BC² = BD² + e²
Since d>e, d² > e², and AB² > BC² or AB > BC
Clearly Sufficient. (In fact again, we didn't have to calculate the exact relation between AB and BC, since we know that d≠e, d²≠e² and consequently AB²≠BC² or AB≠BC)
Hence, either statement alone is sufficient, option D.
Posted from my mobile device
(1) x>y
We know that AB*cos(x) = BC*cos(y) = BD
AB/BC = cos(y)/cos(x) and since y < x, cos(y) > cos(x) or cos(y)/cos(x) > 1 or AB/BC > 1 or AB>BC
Clearly Sufficient. (Infact we didn't have to calculate the exact relation between AB & BC, since we had been given x>y we can infer x≠y and thus cos(x)≠cos(y) and thus AB≠BC)
(2) d > e
We know that AB² = BD² + d² and BC² = BD² + e²
Since d>e, d² > e², and AB² > BC² or AB > BC
Clearly Sufficient. (In fact again, we didn't have to calculate the exact relation between AB and BC, since we know that d≠e, d²≠e² and consequently AB²≠BC² or AB≠BC)
Hence, either statement alone is sufficient, option D.
Posted from my mobile device
04 Feb 2022, 05:00
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Intuitive method
If AB=BC, we know x=y and d=e.
If AB<>BC, we know x<>y and d<>e
So Either (1) or (2) is sufficient.
More robust method
(1)Only
If x>y,
=> -x < -y
=> 90-x < 90-y
=> Angle BAC < Angle BCA
So, in the triangle ABC, BC < AB
Sufficient.
(2) Only
Apply Pythagorean Theorem to two right triangles.
We know AB > BC
Sufficient
Answer is (D).
If AB=BC, we know x=y and d=e.
If AB<>BC, we know x<>y and d<>e
So Either (1) or (2) is sufficient.
More robust method
(1)Only
If x>y,
=> -x < -y
=> 90-x < 90-y
=> Angle BAC < Angle BCA
So, in the triangle ABC, BC < AB
Sufficient.
(2) Only
Apply Pythagorean Theorem to two right triangles.
We know AB > BC
Sufficient
Answer is (D).
