GMAT Data Sufficiency - In the Figure Above, What is the Area of Semicircle DPA?

Question: In the Figure Above, What is the Area of Semicircle DPA?

1. The area of quadrilateral ABCD is 140.
2. The length of the line segment whose endpoints are B and D is 25.

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

‘In the figure above, what is the area of semicircle DPA?’ is a topic for GMAT Quantitative Reasoning. GMAT quantitative reasoning section analyses the candidates' ability to solve mathematical, and quantitative Reasoning problems and interpret graphic data. This section of the GMAT exam comprises 31 questions that need to be completed in 62 minutes. This topic Data Sufficiency question comes with five options and candidates need to choose the one which is correct. This question type in GMAT Quantitative analyses candidates’ quantitative problems and identifies relevance with the data given.

Solution and Explanation 

Approach Solution 1:

It is asked to find the area of DPA

If we look at statement 1 - If angle ABC had been 90 degrees, we could have calculated AD's length.

However, that is untrue. Anything might happen. Therefore, the area of a quadrilateral is useless.

Therefore statement 1 is Insufficient.

If we look at statement 2

Triangle ABD is a right-angled triangle because angle A is right-angled. We can calculate AD, the semicircle's diameter, from that.

Therefore statement 2 is Sufficient.

The answer is B which is Statement 1 is insufficient, but statement 2 is sufficient.

Correct Answer: B

Approach Solution 2:

There is another approach to this question which is fairly simple and uses a value question approach.

To get the area, we must somehow determine the semicircle's diameter or radius.

Looking at statement 1
The area of quadrilateral ABCD is 140.
The area of the square carved out of the quadrilateral ABCD is given by the formula 6*7=112.

Since we already know that the triangle's remaining area will be 140-112 = 28, we can quickly calculate the diameter using the triangle's area formula.

1/2*(b)*7

Thus, the semicircle's diameter will be 16 + 8 = 24.

As a result, we can calculate the semicircle's area.

Statement 1 is insufficient.

Looking at statement 2

The line segment, whose ends are at B and D, is 25 metres long.

The Pythagorean Theorem can be used to calculate the semicircle's diameter. That makes 24.

Therefore, statement 2 is sufficient.

The answer is B which is Statement 1 is insufficient, but statement 2 is sufficient.

Correct Answer: B

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