Is the Area of the Triangle ABC Greater than the Area of the Triangle DEF? GMAT Data Sufficiency

bySayantani Barman Experta en el extranjero

Question: Is the area of the triangle ABC greater than the area of the triangle DEF?

The value of area of ABC is less than that of perimeter of DEF.
Angles of ABC = Angles of DEF

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

“Is the area of the triangle ABC greater than the area of the triangle DEF?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:
Approach Solution 1

Firstly consider S(1),

Let’s pick numbers: if sides of ABC are 1, 1, and \(\sqrt2\) (a half of a square with sides equal to 1), the area equals to 0.5 and t perimeter is 2 + \(\sqrt2\)

The perimeter is much greater than the area of the triangle with these values.

However, if the sides of ABC are 10, 10, and 10\(\sqrt2\) ; then the perimeter is 20 + 10\(\sqrt2\) and the area is 50.

The perimeter is much smaller than the area.

Hence, we can see that this statement is insufficient in itself.

Now, we will take S(2),

All it tells us is that both the triangles are similar or proportionate to each other, but nothing about their size.
Therefore, this statement is also insufficient in itself.
Now combining both the S(1) and S(2), we came to know that they are also insufficient.
Combining the two statements, we still cannot determine whether the triangles have small values of their sides that yield greater perimeters or larger values that yield greater area measurements.

Correct option: E

Approach Solution 2

Fact (1)- Area of ABC < Perimeter of DEF

Consider both ABC and DEF as right angles triangles with 3, 4, 5 combination

Area of ABC (3, 4, 5) = 3 + 4 + 5 = 12

So, is the area of ABC > the area of DEF ? [Is 6 > 6 ?]

Answer is No

Consider both ABC and DEF are equilateral triangles

ABC sides 3, 3, 3 and DEF sides 2, 2, 2

Area of ABC (3, 3, 3) = \(x^2\frac{\sqrt3}{4}\) = 9\(\frac{\sqrt3}{4}\) \(\approx\)4.25

Perimeter of DEF (2, 2, 2) = 2 + 2 + 2 = 6

So, is the area of ABC > the area of DEF ? [Is 9 \(\frac{\sqrt3}{4}\)> 4\(\frac{\sqrt3}{4}\) ?]

Answer is Yes

Fact (2)- Angles of ABC = Angles of DEF

We already proved in Fact 1 that it's sufficient.

Combining the above two facts, we have nothing new.

Hence, correct answer is:

Correct option: E

Approach Solution 3

Statement (1): It is given that Area of ABC < Perimeter of DEF

To maximize the area of ABC, let’s assume that it’s equilateral triangle with side x. the area becomes \(x^2\frac{\sqrt3}{4}\)

To maximize the area given a perimeter, let’s assume that the triangle DEF is also an equilateral triangle with side, y. The area DEF becomes \(y^2\frac{\sqrt3}{4}\)

Since, we don’t know the length of the sides, x and y, the statement is insufficient.

Statement (2): It is given that Angles of ABC = Angles of DEF

This only tells us that the triangles are similar.

Combining both the statements

Since we already assumed that both are equilateral triangles, and therefore similar, statement 2 doesn’t add any additional information. We still don’t know the length of the sides and therefore, the both statements taken together are insufficient.

Correct option: E

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Is the Area of the Triangle ABC Greater than the Area of the Triangle DEF? GMAT Data Sufficiency

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