In the figure above, what is the perimeter of rectangle ABPQ

Target question: What is the perimeter of rectangle ABPQ?

Statement 1: The area of rectangular region ABCD is 3 times the area of rectangular region ABPQ.
Since we aren't given any information regarding ACTUAL lengths, we cannot determine the perimeter of rectangle ABPQ
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The perimeter of rectangle ABCD is 54
There are many possible cases in which ABCD's perimeter is 54, yet the perimeter of rectangle ABPQ is not fixed (see diagrams below)
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several scenarios that satisfy BOTH statements. Here are two:

Case a: The measurements are as follows:

Notice that the area of ABCD (72 square units) is 3 times the area of rectangular region ABPQ (24 square units)
Also notice that ABCD has perimeter 54
In this case, the perimeter of rectangle ABPQ = 1 + 24 + 1 + 24 = 50


Case b: The measurements are as follows:

Notice that the area of ABCD (126 square units) is 3 times the area of rectangular region ABPQ (42 square units)
Also notice that ABCD has perimeter 54
In this case, the perimeter of rectangle ABPQ = 2 + 21 + 2 + 21 = 46

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

C?

My first would have been E, but then I saw AB was common in both rectangles, so I decided to answer C. But I decided to do a little calculation to verify. That took me like three-four minutes. Now seems like C is a good choice.

A...clearly insufficient ..we only know A [ ABPQ] = 1/3 * A[ABCD]
3 * L * W1 = L * W
W = 3 W1
B... clearly insufficient
L + W = 27 ....

combine....
A[ ABPQ] = LW/3

L + W =27 W = 27 - L . W1 = 27 - L / 3

Perimeter of ABPQ = 2 L + 2 (27 - L /3)

I dont see how Wonder got C...what did i miss?

My answer E.

Thanks
praetorian

Let the sides of ABCD be x and y.
Let the sides of ABPQ be x1 and y.

We want to find \(2x1+2y=?\)

1) tells us : \(2x+2y=54\) not sufficient
2) tells us \(xy=3*x1*y\), \(x=3*x1\) not sufficient

1+2) tells us \(6x1+2y=54\), so \(3x1+y=27\)

x1 could be equal to 1 and y to 24
or x1 could be equal to 2 and y 21 etc.

Not sufficient, answer E

I read in the Manhattan Geometry Guide (13 ed.)
"If two similar triangles have corresponding side lengths in ratio a: b, then their
areas will be in ratio a^2: b^2.
The lengths being compared do not have to be sides— they can represent heights or perimeters. In fact,
the figures do not have to be triangles."

So I guess, the parameters in this case should be in ratio as well.
Can some one explain this to me.

Hi,
you are correct --
If two polygons are similar, their corresponding sides, altitudes, medians, diagonals, angle bisectors and perimeters are all in the same ratio.

but here "You are not given anything about similarity", so you cannot apply the same rule here..
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Thanks, but I am finding it difficult to figure out why ABPQ and ABCD are not similar, since both the figures are rectangle, they have one common side therefore PQ has to be || with CD. is there anything wrong with my concept??

Hi,
since you are also saying one side is equal and the other two side are lesser than their corresponding sides ..
this basically means the sides are not in the same ratio..
whereas ONE set of sides have ratio as 1, the OTHER set is not 1, but something else..
for two triangles - ONE has L= 2 and B= 1, th eOTHER has L=6 and B=3.. these triangles are similar as ratios of sides 2/6 is same as 1/3
BUT if ONE has L= 2 and B= 1, th eOTHER has L=6 and B=1.. these triangles are NOT similar as ratios of sides 2/6 is NOT same as 1/1
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Got you point Chetan. Thanks for the clarification.

Lets dive into statement analysis straight away:

Stmt 1: Since AB is common width between 2 figures, clearly the length of larger rectangle ABCD should be 3 times the length of smaller rectangle ABPQ. But using this alone we dont have concrete values to work with. Hence (1) is insufficient.
Stmt 2: Perimeter of ABCD = 54.
= > 2(l+b) =54
=> l+b = 27. This does not give any clue about the smaller rectangle. Hence (2) is insufficient.

Using (1) & (2)
=> 3l' + b = 27. (where l' is length of ABPQ)
We can't solve this further to get 2(l'+b). Hence E.
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I notice that this is quite a common trick question on areas/perimeters that and can be solved very quickly.

Basically for rectangles, the perimeter and area may not be proportionate to each other (do correct me if I am wrong)

For example a 100 sqft rectangle could be:
1) 100 + 100 + 1 +1 = 202 ft perimeter
2) 20 + 20 + 5 + 5 = 50 ft perimeter

Back to the question,
(1) The area ratio is given, but nothing about the perimeter. Also, the question is asking for a specific value which the statement does not provide any clue - Insufficient
(2) Although the perimeter of ABCD is given, it does not show how ABCD and ABPQ are related to each other - Insufficient
(1) & (2) Since area and perimeter are not proportionate, not possible to derive answer without also knowing any side lengths - Insufficient

Hence answer is (E)

If the question was on a square, then it would have been (C) since we could derive the length of its sides from statement (2)

The perimeter of ABPQ = 2 * (AB + AQ)
THe first statement talks about only ratios and no absolute value has been given.
So First statement is not sufficient.

The second statement says , 2 * (AB + AD) = 54 ; AB + AD = 27
It does not provide any value of AQ . So we cant calculate the value of 2*(AB + AQ) .Hence statement 2 is also not sufficient.

Taken together,
From statement 1 ,
3 *(AQ *AB) = AB *AD
3*AQ = AD

Now ,
2*(AB + AQ ) = 2 * (AB + AD/3) = 2 * ((3*AB + AD) /3) = 2 * ((2*AB +27)/3) = 4/3 * AB + 18
We dont know the value of AB. Hence we cant find the perimeter.
Option E is the answer.
Please give me KUDO s if you liked my answer.

I did this one algebraically. Let's label the smallest side BP as b, PC as a and AB as c. So three sides: a,b,c

a) (a + b) x c = 2(bc) <---bc is the area of the smaller rectangle we are looking for
Clearly insufficient

b) 2(a + b) + 2c = 54
a + b + c = 27
Clearly insufficient

C: 3 unknowns, 2 equations

E.
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The way I thought about this question is:

Because height is the same. Width of smaller = 1/3 width of larger

We're looking for
2h + 2/3*w=perimeter of smaller

We know
2h+2w=54

Because we don't know how the perimeter of larger is distributed between width and length, we don't have sufficient information to know perimeter of smaller.

Algebraically

Subtract first from second
4/3 * w = 54 - perimeter of smaller

Solve
perimeter of smaller = 54 - 4/3 * w

We have no more information on w so insufficient

Question Stem Analysis:

We need to determine the perimeter of the rectangle ABPQ. We know the rectangles ABPQ and ABCD share a common side, namely AB. Let AB = x, BP = y, and PC = z. Then the area of ABPQ is xy, and the area of ABCD is x(y + z). The perimeter of ABPQ is 2x + 2y, and the perimeter of ABCD is 2x + 2(y + z) = 2x + 2y + 2z.

Statement One Alone:

\(\Rightarrow\) The area of rectangular region ABCD is 3 times the area of rectangular region ABPQ.

Since we are told that the area of ABCD is 3 times the area of ABPQ, we can write:

\(\Rightarrow\) a(ABCD) = 3 * a(ABPQ)

\(\Rightarrow\) x(y + z) = 3 * xy

\(\Rightarrow\) y + z = 3y

\(\Rightarrow\) z = 2y

We were able to determine that the length of PC is twice the length of BP, however, this is not sufficient to determine an answer to the question.

If x = 1, y = 1, and z = 2, then the area of ABPQ is 1, and the area of ABCD is 3. We see that the condition that the area of ABCD is three times the area of ABPQ is satisfied. In this case, the perimeter of ABPQ is 4.

If x = 2, y = 1, and z = 2, then the area of ABPQ is 2, and the area of ABCD is 6. Once again the condition that the area of ABCD is three times the area of ABPQ is satisfied. In this case, the perimeter of ABPQ is 6.

Statement one alone is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

\(\Rightarrow\) The perimeter of rectangle ABCD is 54.

Thus, 2x + 2y + 2z = 54, which means that x + y + z = 27. This is not sufficient to determine the perimeter of ABPQ, as we can find many triples x, y, z adding up to 27. For instance, if x = 1, y = 1, and z = 25, then the perimeter of ABPQ is 4. On the other hand, if x = 1, y = 2, and z = 24, then the perimeter of ABPQ is 6.

Statement two alone is not sufficient. Eliminate answer choice B.

Statements One and Two Together:

We have the equations x + y + z = 27, and z = 2y. Substituting z = 2y in the first equation, we get x + 3y = 27. Since we have two unknowns but only one equation, this is not sufficient to determine a unique value for 2x + 2y.

If x = 3 and y = 8, then the perimeter of ABPQ is 22. If x = 6 and y = 7, then the perimeter of ABPQ is 26. I leave it as an exercise to show that in both of these cases, the area of ABCD is three times the area of ABPQ, and the perimeter of ABCD is 54. Statements one and two together are not sufficient.

Answer: E
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