Figure ABCD is a Rectangle with Sides of Length x Centimeters and Width y Centimeters, and a Diagonal of Length z Centimeters.

Question: Figure ABCD is a rectangle with sides of length x centimeters and width y centimeters, and a diagonal of length z centimeters. What is the measure, in centimeters, of the perimeter of ABCD ?

(1) x – y = 7
(2) z = 13

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

“Figure ABCD is a rectangle with sides of length x centimeters and width y centimeters, and a diagonal of length z centimeters” – 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.

Solution and Explanation:

Approach 1:

Given:

• ABCD is a rectangle
• length x centimeters
• width y centimeters
• diagonal of length z centimeters

Find Out:

• The perimeter of ABCD ?

Since ABCD is a rectangle, so it is made of two right triangle.
Hence, we know that the equation is: x^2 + y^2 = z^2.

We need to know the perimeter of ABCD.
The formula for the perimeter is 2(x+y).
Now, let us consider each statement one by one:

Statement 1:
x - y = 7, but we need the value of x +y.
Since we cannot determine any value, hence it is not sufficient.

Statement 2:
z = 13 or, z^2 = 169,
=>x^2 + y^2 =169,
=>(x+y)^2 -2xy =169.
We see that we cant find (x+y) from it.
Hence, the second statement is also not sufficient.

Now we will combine both the statements:
169 = x^2 + (x- 7)^2 or
=> 2x^2 -14x+49 =169 or
=> 2x^2 -14x -120 =0, or
=> x^2 -7x -60 =0,
=> x^2 -12x +5x -60 =0.

So x is 12 and y is 5. The statement is Sufficient.

Hence, C is the answer.

Approach 2:

Given:

• ABCD is a rectangle
• length x centimeters
• width y centimeters
• diagonal of length z centimeters

Find Out:

• The perimeter of ABCD ?

If we look at z=13 as
x^2+y^2=z^2=169
(x+y)(x-y)=169
(x+y)=169/7..... as from 1.

Now, we know that x-y=7

Hence perimeter is 2(x+y)
= 2(169/7)

Since both the statements together are sufficient, C is the correct answer

Approach 3:

In the question, since we are not told that x and y are integers, we will check each one by one:

We have statement 1:
x-y=7
thus x^2+y^2 -2xy=49. so not sufficient.

Statement 2 alone:
z=13, well, z^2 = x^2 + y^2.
we have 169=x^2 + y^2.
not sufficient.
Now, we need to find the perimeter, or 2(x+y).

we are told that x^2+y^2-2xy = 49

now we have
169=x^2 + y^2.
combine both:
169-2xy=49
120=2xy
60=xy.
now:
xy=60
x^2+y^2=169.

combine both:
x^2+y^2 + 2xy = 169+120
(x+y)^2 = 289.

now, this is sufficient, as we can find x+y, and thus find 2(x+y).

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