Is the Perimeter of a Rectangle Greater than 8 inches GMAT Data Sufficiency

Question: Is the perimeter of a rectangle greater than 8 inches?

1. The diagonal of the rectangle is twice as long as it shorter side
2. The diagonal of the rectangle is 4 inches longer than its shorter side

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: B

Approach Solution (1):

The question asks whether Perimeter = 2 (a+b) > 8, or whether a + b > 4, (where a and b are the length of the sides of the triangle).

(1) The diagonal of the rectangle is twice as long as its shorter side. Clearly insufficient, we know the shape of the rectangle but not its size.
(2) The diagonal of the rectangle is 4 inches longer than its shorter side. This statement basically says that the length of the diagonal is greater than 4 inches: d > 4. Now, consider the triangle made by the diagonal and the two sides of the rectangle: since the length of any side of a triangle must be smaller than the sum of the other two sides, then we have that a + b > d, so a + b > d > 4.

Sufficient

Approach Solution (2):

(1) Diagonal = 2 times short

Suppose short = 4, then perimeter > 2 * 4 = 8, YES

Suppose short = 1, diagonal = 2, long = root (4 – 1) = root (3) and perimeter = 2 root3 + 2 = approx 3.4 + 2 < 8

Not sufficient

(2) The diagonal is 4 inches longer than short.

Label the short side x and the long side y. \(x^2+y^2=(x+4)^2\)and \(x^2+y^2=x^2+8x+16\), and \(y^2=8x+16\)and y = root (8x + 16) so min y > 4, min y > 4, so min perimeter > 8

So sufficient.

“Is the perimeter of a rectangle greater than 8 inches?”- 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. GMAT Quant practice papers help the candidates to analyze varieties of types of questions that will brush up their mathematical learning.

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