Sides AB, BC, and CD of quadrilateral ABCD all have length 10. What is

The question stem is asking us if we can find a definite area of quadrilateral ABCD, to do this we must know the exact shape of the quadrilateral including angles and appropriate side lengths.

1) This information tells us that it could be a square but without knowing that the side AD is also equal to 10 we cannot eliminate the possibility of the quadrilateral being a trapezoid which we would be unable to calculate the area of without further information regarding the shape. Not Sufficient eliminate answer choices A and D.

2) We still do not have information that would allow us to eliminate the possiblity of the shape being a trapezoid as we do not have the other diagonal that we would need to determine all angles and side lengths for the quadrilateral. Not Sufficient eliminate answer choice B.

1+2) Together we still do not have enough information without knowing more angles and lengths to determine the shape or respective area of the quadrilateral. Not Sufficient

Select answer choice E.

(Ans: E)
we can not specify the what type of quadrilateral it is so Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data is required to ans the question.

MANHATTAN GMAT OFFICIAL SOLUTION:

We are told that three of the sides of quadrilateral ABCD have length 10. We are told nothing else about that quadrilateral, and we are asked for its area.

A good way to visualize the constraints is to use “rubber-band geometry.” Picture the three sides that we do know as stiff rods, each of length 10. They are hooked together with flexible hinges, because we know nothing about the angles in the polygon. Finally, the fourth side (AD) is a rubber band – it can stretch and shrink as we flex the hinged rods.


Statement (1): BC is parallel to AD.

Fiddle with the mental contraption of rods and hinges. To make rod BC parallel to the rubber band, you can make either a trapezoid or a parallelogram. This is because the two other sides (AB and CD) have the same length (= 10). The parallelogram will in fact be a rhombus: AD will also have length 10.


Even if you knew which figure was in play, you don’t have a fixed height, so you don’t know the area.

Statement (1) is NOT SUFFICIENT.

Statement (2): Diagonal AC,which lies inside the quadrilateral, has length 10√3.

This statement tells us that triangle ABC is fixed, and it looks something like this:


We don’t need to know the exact angles; all we need to know is that the triangle is fixed in position (and thus has a fixed area). However, CD can swing through any number of angles, making the area of the other triangle (ACD) variable.

Statement (2) is NOT SUFFICIENT.

Statements (1) and (2) TOGETHER: We get the following pictures:


Notice that either picture is possible, as long as the length of the diagonal is longer than that of a square with sides 10. That diagonal would have length 10√2. Since 10√3 is longer, we have the two possibilities above. The trapezoid actually contains the rhombus, so it has a larger area.

Even together, the statements are not sufficient.

The correct answer is E.

Sir,
Why did not you consider a square in statement A.

we can consider Square also, But that's doesn't matter because it will not be unique answer.
it can be square that there is not always a square.

Hence insufficient.

We are provided with the length of 3 sides.

From S1: We do not know what type of quad it is...hence there can be multiples ans. Not Sufficient

From S2: We are told that the diagonal length, but thaat in itself is not sufficient for a unique area. Not Sufficient.

Combining, we are not able get any definite quad area. Hence, both are not sufficient.

Ans is (E).