What is the Perimeter of Rectangle ABCD GMAT Data Sufficiency

Question: What is the perimeter of rectangle ABCD?

(1) Diagonal BD has length 10
(2) BDC has measure 30 degree

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.

Correct Answer: C

Solution and Explanation:
Approach Solution 1:

There is only one approach to this problem.

As per the formula, the perimeter of the rectangle ABCD is 2(AB + BC).

Statement (1) alone: Diagonal BD has length = 10 units.
As we all know that every angle in a rectangle is a right angle. Therefore, we have a right-angled triangle BDA with BD, the hypotenuse, being 10 units in length.

If BD = 10, then AB = 6 and AD = 8.
Therefore, the perimeter of rectangle = 2(6+8) = 2(14) = 28.

If BD = 10, AB = 5√2 and AD = 5√2.
Therefore, the perimeter of rectangle = 2(5√2+ 5√2 ) = 20√2.

Hence, statement (1) alone is not insufficient to get a unique value for the perimeter of the rectangle.

Statement (2) alone: ∠BDC = 30 degrees.

This implies that ∠CBD = 60 degrees since ∠BCD = 90 degrees.
This indicates that the rectangle is divided into two “30-60-90” right-angled triangles. This signifies that the sides opposite to these angles i.e. BC, CD and BD are in the ratio of 1: √3 : 2.
However, by knowing the ratio of the sides, the sides of the rectangle cannot be known.

Therefore, statement (2) alone is not sufficient to get the lengths of the sides.
Hence the perimeter of the rectangle ABCD cannot be derived.

Combining statements (1) and (2), we get:

Statement (2) tells us that the rectangle has been divided into two “30-60-90” right-angled triangles. We also derive that BC : CD : BD = 1: √3 : 2.
From statement (1) alone, we get that BD = 10 unit.

If BD = 10, BC = 5 units and CD = 5√3 units.

Since ABCD is a rectangle, CD = AB.
Therefore, perimeter of rectangle = 2(AB + BC) = 2(5√3 + 5) units.

Both statements together are sufficient to find the perimeter of the rectangle.

“What is the perimeter of rectangle ABCD”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2019”. The GMAT Quant section comprises a total of 31 questions. GMAT Data Sufficiency questions are followed by a problem statement and two factual statements. GMAT data sufficiency comes up with 15 questions which are two-fifths of the total 31 GMAT quant questions.

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