## What is the perimeter of rectangle ABCD

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### What is the perimeter of rectangle ABCD

by BTGmoderatorDC » Wed Nov 15, 2017 2:11 pm
What is the perimeter of rectangle ABCD?

(1) The longer side of the rectangle is 2 meters shorter than its diagonal
$$(2)The\ ratio\ of\ the\ shorter\ side\ of\ the\ rec\tan gle\ to\ its\ diagonal\ is\ \frac{1}{3}\$$

Which of the statement is sufficient?

OA C

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by Rich.C@EMPOWERgmat.com » Wed Nov 15, 2017 8:58 pm
Hi lheiannie07,

We're asked for the perimeter of rectangle ABCD. To answer this question, we'll need to figure out the Length (L) and Width (W) of the rectangle. This prompt involves a great 'System Algebra' shortcut that you can use to avoid doing a lot of math.

1) The longer side of the rectangle is 2 meters shorter than its diagonal

With the information in Fact 1, we can create 2 equations (with the diagonal represented by D):
L = D - 2
L^2 + W^2 = D^2

Unfortunately, with 3 unknowns, but only 2 equations, we cannot solve for any of the variables, so we cannot figure out the perimeter.
Fact 1 is INSUFFICIENT

2) The ratio of the shorter side to the diagonal is 1/3

With the information in Fact 2, we can create 2 equations (one of which is the same as we created in Fact 1):
W/D = 1/3....... D = 3W
L^2 + W^2 = D2

Again, with 3 unknowns, but only 2 equations, we cannot solve for any of the variables, so we cannot figure out the perimeter.
Fact 2 is INSUFFICIENT

Combined, we know that we're dealing with a rectangle, so the variables can ONLY be POSITIVE numbers. As such, even though we're dealing with 'squared terms', the negative answers are not possible here. We have 3 variables and 3 unique equations, so we CAN solve for all 3 variables - and there will be just one solution.
Combined, SUFFICIENT

GMAT assassins aren't born, they're made,
Rich
Contact Rich at Rich.C@empowergmat.com

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by BTGmoderatorDC » Wed Jan 10, 2018 10:01 pm
Rich.C@EMPOWERgmat.com wrote:Hi lheiannie07,

We're asked for the perimeter of rectangle ABCD. To answer this question, we'll need to figure out the Length (L) and Width (W) of the rectangle. This prompt involves a great 'System Algebra' shortcut that you can use to avoid doing a lot of math.

1) The longer side of the rectangle is 2 meters shorter than its diagonal

With the information in Fact 1, we can create 2 equations (with the diagonal represented by D):
L = D - 2
L^2 + W^2 = D^2

Unfortunately, with 3 unknowns, but only 2 equations, we cannot solve for any of the variables, so we cannot figure out the perimeter.
Fact 1 is INSUFFICIENT

2) The ratio of the shorter side to the diagonal is 1/3

With the information in Fact 2, we can create 2 equations (one of which is the same as we created in Fact 1):
W/D = 1/3....... D = 3W
L^2 + W^2 = D2

Again, with 3 unknowns, but only 2 equations, we cannot solve for any of the variables, so we cannot figure out the perimeter.
Fact 2 is INSUFFICIENT

Combined, we know that we're dealing with a rectangle, so the variables can ONLY be POSITIVE numbers. As such, even though we're dealing with 'squared terms', the negative answers are not possible here. We have 3 variables and 3 unique equations, so we CAN solve for all 3 variables - and there will be just one solution.
Combined, SUFFICIENT

GMAT assassins aren't born, they're made,
Rich
Thanks a lot!

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Posts: 5054
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by BTGmoderatorDC » Sun Jan 14, 2018 4:53 pm
Rich.C@EMPOWERgmat.com wrote:Hi lheiannie07,

We're asked for the perimeter of rectangle ABCD. To answer this question, we'll need to figure out the Length (L) and Width (W) of the rectangle. This prompt involves a great 'System Algebra' shortcut that you can use to avoid doing a lot of math.

1) The longer side of the rectangle is 2 meters shorter than its diagonal

With the information in Fact 1, we can create 2 equations (with the diagonal represented by D):
L = D - 2
L^2 + W^2 = D^2

Unfortunately, with 3 unknowns, but only 2 equations, we cannot solve for any of the variables, so we cannot figure out the perimeter.
Fact 1 is INSUFFICIENT

2) The ratio of the shorter side to the diagonal is 1/3

With the information in Fact 2, we can create 2 equations (one of which is the same as we created in Fact 1):
W/D = 1/3....... D = 3W
L^2 + W^2 = D2

Again, with 3 unknowns, but only 2 equations, we cannot solve for any of the variables, so we cannot figure out the perimeter.
Fact 2 is INSUFFICIENT

Combined, we know that we're dealing with a rectangle, so the variables can ONLY be POSITIVE numbers. As such, even though we're dealing with 'squared terms', the negative answers are not possible here. We have 3 variables and 3 unique equations, so we CAN solve for all 3 variables - and there will be just one solution.
Combined, SUFFICIENT

GMAT assassins aren't born, they're made,
Rich
Thanks a lot!

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by Scott@TargetTestPrep » Thu Jan 18, 2018 8:02 am
lheiannie07 wrote:What is the perimeter of rectangle ABCD?

(1) The longer side of the rectangle is 2 meters shorter than its diagonal
$$(2)The\ ratio\ of\ the\ shorter\ side\ of\ the\ rec\tan gle\ to\ its\ diagonal\ is\ \frac{1}{3}\$$
We need to determine the perimeter of rectangle ABCD.

Statement One Alone:

The longer side of the rectangle is 2 meters shorter than its diagonal.

We can let the length of the longer side = L and the length of the diagonal = G. Thus we have L = G - 2. However, since we don't know the value of L or G, we can't determine the perimeter of ABCD. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

The ratio of the shorter side of the rectangle to its diagonal is 1/3.

We can let the length of the shorter side = W and the length of the diagonal = G. Thus we have W = G/3. However, since we don't know the value of W or G, we can't determine the perimeter of ABCD. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the two statements, we see that L = G - 2 and W = G/3. Notice that the perimeter of ABCD is 2L + 2W. Furthermore, L, W and G form a right triangle with G as the hypotenuse. Thus, we have:

L^2 + W^2 = G^2

Now, substituting G - 2 for L and G/3 for W, we have:

(G - 2)^2 + (G/3)^2 = G^2

From the above equation, we can solve for G. Once we solve for G, we can determine the values of L and W and, hence, we can determine the perimeter of ABCD.