In triangle ABC, angle ABC is 105 degrees. Is the area

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Sat Sep 29, 2018 2:40 am

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In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?

(1) Segment AB = 2âˆš2
(2) Angle BCA is 30 degrees.

OA C

Source: Veritas Prep

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Source: Beat The GMAT — Data Sufficiency | Sat Sep 29, 2018 2:40 am

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Sun Sep 30, 2018 10:54 pm

BTGmoderatorDC wrote:In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?

(1) Segment AB = 2âˆš2
(2) Angle BCA is 30 degrees.

OA C

Source: Veritas Prep

None of the statements alone is sufficient to answer the question, so we combine them.

See the image with construction.

Drop a perpendicular from B to AC; it meets at point D. /_BDC = /_BDA = 90Âº; /_CBD = 60Âº; /_BAD = 45Âº

We have to get the area of âˆ†ABC = 1/2 * BD * AC = BD(AD + DC)/2

1. Given âˆ†ABD, a 45-45-90 triangle and AB = 2âˆš2, we have BD = AD = (2âˆš2)/âˆš2 = 2
2. Given âˆ†BDC, a 30-60-90 triangle and BD = 2, we have CD = 2âˆš3

Thus, area of âˆ†ABC = BD(AD + DC)/2 = 2(2 + 2âˆš3)/2 = 2 + 2âˆš3 = A unique value

Though the value of 2 + 2âˆš3 can be calculated, it is not required to be computed. There can be three scenarios.

1. 2 + 2âˆš3 = 5. In this case, the answer is No.
2. 2 + 2âˆš3 > 5. In this case, the answer is No.
3. 2 + 2âˆš3 < 5. In this case, the answer is Yes.

Note that since 2 + 2âˆš3 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: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Oct 01, 2018 11:22 am

BTGmoderatorDC wrote:In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?

(1) Segment AB = 2âˆš2
(2) Angle BCA is 30 degrees.
Source: Veritas Prep

Nice arguments, Jay!

LetÂ´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)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Mon Oct 01, 2018 11:50 am

$$Question\ =\ Is\ the\ area\ of\ triangle\ ABC\ \angle5$$
$$I.e\ is\ \frac{1}{2}\cdot b\cdot h\ \angle\ 5\ or\ is\ \frac{bh}{2}\ \angle\ 5?$$

$$Statement\ 1\ =\ Segment\ AB\ =\ 2\sqrt{2}$$
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.

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