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06 Nov 2009, 05:58
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(1) \(BD=6\sqrt{3}\)
(2) x = 60
06 Nov 2009, 20:26
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yangsta8
So just for my knowledge, if we know 2 sides of a triangle, and the angle in between we can safely determine that the information is sufficient?
This is a very good question. Well, I think everybody agrees that knowing such tips is very important for GMAT. Especially in DS as it helps to avoid time wasting by not calculating an exact numerical values.
When can we say that information given is sufficient to calculate some unknown value in triangle? Think it's the same as determining congruency. If we are given some data and we can conclude that ONLY one triangle with given measurements exists, it should mean also that with given data we can calculate anything regarding this triangle.
Determining congruency:
1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.
So, knowing SAS or ASA is sufficient to determine unknown angles or sides.
NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.
Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.
The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.
To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.
So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.
Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.
So, knowing three angles is NOT sufficient to determine lengths of the sides.
In our original question we had had SAS situation with (1), and ASA situation in (2) so each alone was indeed sufficient to calculate any other unknown value in this triangle.
06 Nov 2009, 10:55
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Draw BE perpendicular to AD, let \(CE = y\).
stmt1:
\(sin30 = BE/BD\), BD is known, so BE can be found as sin30 =1/2.
\(cos30 = DE/BD\), \(\sqrt{3}/2 = (6+y)/BD\) , so y can be found.
Now we know BE and CE, apply pythagoras and find BC.
stmt2:
\(sin30 = BE/BD\), BD is known, so BE can be found as sin30 =1/2.
Now apply \(sin60 = BE/BC\), so BC can be found.
D
stmt1:
\(sin30 = BE/BD\), BD is known, so BE can be found as sin30 =1/2.
\(cos30 = DE/BD\), \(\sqrt{3}/2 = (6+y)/BD\) , so y can be found.
Now we know BE and CE, apply pythagoras and find BC.
stmt2:
\(sin30 = BE/BD\), BD is known, so BE can be found as sin30 =1/2.
Now apply \(sin60 = BE/BC\), so BC can be found.
D
