In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?
(1) Segment AB = 2√2
(2) Angle BCA is 30 degrees.
OA C
Source: Veritas Prep
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In triangle ABC, angle ABC is 105 degrees. Is the area
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BTGmoderatorDC
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Jay@ManhattanReview
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None of the statements alone is sufficient to answer the question, so we combine them.BTGmoderatorDC wrote:In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?
(1) Segment AB = 2√2
(2) Angle BCA is 30 degrees.
OA C
Source: Veritas Prep
See the image with construction.
Drop a perpendicular from B to AC; it meets at point D. /_BDC = /_BDA = 90º; /_CBD = 60º; /_BAD = 45º
We have to get the area of ∆ABC = 1/2 * BD * AC = BD(AD + DC)/2
1. Given ∆ABD, a 45-45-90 triangle and AB = 2√2, we have BD = AD = (2√2)/√2 = 2
2. Given ∆BDC, a 30-60-90 triangle and BD = 2, we have CD = 2√3
Thus, area of ∆ABC = BD(AD + DC)/2 = 2(2 + 2√3)/2 = 2 + 2√3 = A unique value
Though the value of 2 + 2√3 can be calculated, it is not required to be computed. There can be three scenarios.
1. 2 + 2√3 = 5. In this case, the answer is No.
2. 2 + 2√3 > 5. In this case, the answer is No.
3. 2 + 2√3 < 5. In this case, the answer is Yes.
Note that since 2 + 2√3 is a unique value, only one of the three scenarios would occur and we would get a unique answer. Sufficient.
The correct answer: C
Hope this helps!
-Jay
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fskilnik@GMATH
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Nice arguments, Jay!BTGmoderatorDC wrote:In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?
(1) Segment AB = 2√2
(2) Angle BCA is 30 degrees.
Source: Veritas Prep
Let´s see the problem in "another angle"! (I know, bad joke...)
\[\angle ABC = {105^ \circ }\,\,\]
\[{S_{\Delta ABC}}\,\,\mathop < \limits^? \,\,\,5\]
(2) We have two (therefore all three) internal angles of triangle ABC.
This is not enough:
(1+2) We "know" the triangle ABC! I mean, all its angles, all its side lengths.
Reason: any two triangles that satisfy the question stem (pre-statements) and statement (2) are similar.
When we add statement (1) info, we are able to find the "scale" involved, hence any two triangles that satisfy
the question stem AND both statements together are CONGRUENT. In a sense, there is just one triangle ABC!
Conclusion: the area of triangle ABC is UNIQUE, hence our answer is also UNIQUE. Sufficient!
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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deloitte247
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$$Question\ =\ Is\ the\ area\ of\ triangle\ ABC\ \angle5$$
$$I.e\ is\ \frac{1}{2}\cdot b\cdot h\ \angle\ 5\ or\ is\ \frac{bh}{2}\ \angle\ 5?$$
$$Statement\ 1\ =\ Segment\ AB\ =\ 2\sqrt{2}$$
This only provides us with the length of side AB and it is not enough to find the area of triangle ABC.
Hence, Statement 1 is NOT SUFFICIENT.
$$Statement\ 2=\ Angle\ BCA\ =\ 30^0$$
This only provides us with the value of angle C and it is not enough to find the area of triangle ABC.
Hence, Statement 2 is NOT SUFFICIENT.
STATEMENT 1 and 2 together;
$$From\ statement\ 1\ we\ deduce\ that\ AB\ =\ 2\sqrt{2}$$, if we split triangle ABC into two triangles by drawing a line from the angle ABC we can get two right - angle triangles, the first triangle will have angles 90^0, 30^0, and 60^0.
From this, we can use SOH, CAH, TOA formula to find the remaining sides and all the needed values to find the area of the triangle.
Hence, Statement 1 and 2 together is SUFFICIENT.
Option C is CORRECT.
$$I.e\ is\ \frac{1}{2}\cdot b\cdot h\ \angle\ 5\ or\ is\ \frac{bh}{2}\ \angle\ 5?$$
$$Statement\ 1\ =\ Segment\ AB\ =\ 2\sqrt{2}$$
This only provides us with the length of side AB and it is not enough to find the area of triangle ABC.
Hence, Statement 1 is NOT SUFFICIENT.
$$Statement\ 2=\ Angle\ BCA\ =\ 30^0$$
This only provides us with the value of angle C and it is not enough to find the area of triangle ABC.
Hence, Statement 2 is NOT SUFFICIENT.
STATEMENT 1 and 2 together;
$$From\ statement\ 1\ we\ deduce\ that\ AB\ =\ 2\sqrt{2}$$, if we split triangle ABC into two triangles by drawing a line from the angle ABC we can get two right - angle triangles, the first triangle will have angles 90^0, 30^0, and 60^0.
From this, we can use SOH, CAH, TOA formula to find the remaining sides and all the needed values to find the area of the triangle.
Hence, Statement 1 and 2 together is SUFFICIENT.
Option C is CORRECT.
