Data Sufficiency

I.

Is quadrilateral ABCD a rectangle?

(1) Line segments AC and BD bisect one another.

(2) Angle ABC is a right angle.

II.

Is quadrilateral ABCD a rhombus?

(1) Line segments AC and BD are perpendicular bisectors of each other.

(2) AB = BC = CD = AD

III.

Is quadrilateral ABCD a square?

(1) ABCD is a rectangle.

(2) AB = BC


Please help.

sumit.sinha wrote:I.

Is quadrilateral ABCD a rectangle?

(1) Line segments AC and BD bisect one another.

(2) Angle ABC is a right angle.

(1) Line segments AC and BD bisect one another.
A quadrilateral whose diagonals bisect each other can be a Parallelogram or Rhombus or Rectangle or Square.
Parallelogram and Rhombus belong to Parallelogram family.
Rectangle and Square belong to Rectangle family.
Hence we do not know if the quadrilateral is Rectangle or Parallelogram.
INSUFF

(2) Angle ABC is a right angle.
Based on only one angle we cant determie the shape of quadrilateral.
INSUFF

Combining:
A parallelogram with one of the angles as 90 degrees is a rectangle.
hence quadrilateral is a rectangle.

pick C.

Hope this helps!!

sumit.sinha wrote: II.

Is quadrilateral ABCD a rhombus?

(1) Line segments AC and BD are perpendicular bisectors of each other.

(2) AB = BC = CD = AD

(1) Line segments AC and BD are perpendicular bisectors of each other.

Diagonals are perpendicular bisectors in a rhombus or square.
square is a rhombus with each angle as 90.
Hence SUFF

(2) AB = BC = CD = AD
A quadrilateral with all four sides equal is called a rhombus.
Hence SUFF

pick D.

Hope this helps!!

sumit.sinha wrote:I.
III.
Is quadrilateral ABCD a square?
(1) ABCD is a rectangle.
(2) AB = BC

(1) ABCD is a rectangle.
A rectangle cannot be considered a square unless all sides are equal. But the other way round is true.
Hence INSUFF

(2) AB = BC
if 2 adjacent sides are equal, it doesnt guarantee a square.
INSUFF

Combining:
We know that the opposite sides of rectangle are equal. And we have that the adjacent sides are also equal from stmt2.
hence all sides are equal.
Therefore the quad is square.

pick C.

Hope this helps!!

II.
Is quadrilateral ABCD a rhombus?
(1) Line segments AC and BD are perpendicular bisectors of each other.

(2) AB = BC = CD = AD

@kvcpk..How is statement1 sufficient? can you pls explain me.. I'm getting confused..

(1)Line segments AC and BD are perpendicular bisectors of each other>every parallelogram family has that property
i guess.

Parallelogram.
properties:*diagonals bisect each other
*opp sides equal
*opp angles equal
divided into (two types) Rectangle and Rhombus

Rectangle....................................................................Rhombus
properties:*inherit those of Parallelogram+...............prop:*inherit from above Parallelogram+
*angles are 90degrees .............................................*all sides are equal

combining both the properties.
>square.
*all angels 90 and all sides equal

>> "I am thinking answer is B"
statement1>> it says it is a paralleogram
statement2>> it says a rhombus.. it can be a square but square is also a rhombus!

My answers:

Q1.
statement1>parallelogram
statement2>rectangle only if statement1 is true
answer:"C"
Q2.
answer:"B"
Q3
statement1>rectangle meaning all angles are 90
statement2> rhombus...
together it becomes a square..answer:"C"

Ans 1. Necessary condition for a quadrilateral to be a rectangle is its diagonals bisect each other and angles are of 90 degree.
Statement 1. We are given that the diagonals bisect each other but we are unaware of the angles. From statement 1 we can deduce that the quadrilateral could be a square or a rectangle or a rhombus. Hence, Inusfficient.
Statement 2. Any quadrilateral can have one of its angle equal to 90 degrees. Hence, Insufficient.
Statement 1 & 2 together. Combining the results of statement 1 & 2, we get,
A quadrilateral whose diagonals bisect each other and 1 angle is 90 degree. So, it is definitely a rectangle because when diagonals bisect each other and 1 angles is of 90 degree, then all other angles are also of 90 degree.
Hence, the given quadrilateral is a square or a rectangle.
We know, every square is a rectangle. Hence, the given quadrilateral is definitely a rectangle. Hence, sufficient.

Ans 3. Statement 1. ABCD is a rectangle but we don't know if the sides of this rectangle are equal or not. Hence, Insufficient.
Statement 2. Any quadrilateral can have its two adjacent sides equal. So, we can't surely say that it is a square. Hence, Insufficient.
Statement 1 & 2 together. ABCD is a rectangle and two adjacent sides are equal.
This is only possible if the rectangle is a square. Hence, sufficient.