Geometry

[Koala]

If ABCD is quadrilateral, is AB=BC=CD=DA?

1) AC is perpendicular to BD
2) AB+CD=BC+AD

OA is E

[N:Dure]

He wants to prove that ABCD is a square with sides AB BC CD DA all equal.

Info. 1 tells us that the diagonals intersect and they have 90 degrees together, which doesn't really help.

2 tells us that the addition of the 2 opposite sides are equal. if we assume that AB+CD=BC+AD= 6 AB & CD, BC & AD could have different combinations like (4,2) (3,3) (2,4) (5,1) ..etc so we can't prove that they're all equal.

Thus there isn't sufficient info to prove that this is a square.

[shovan85]

If ABCD is quadrilateral, is AB=BC=CD=DA?

1) AC is perpendicular to BD
2) AB+CD=BC+AD

OA is E

1. AC is perpendicular to BD

Think about the geometrical shapes (as quadrilateral) that has perpendicularly intersecting diagonals.

Being a quadrilateral a square, or a Rhombus, or a Kite can have perpendicular. Though square and rhombus has all the sides equal, Kite does not have equal sides.

Thus insufficient.

2. AB+CD=BC+AD

Take simple numbers to verify.

3+4 = 2+5 (=7) so does this mean that 3 = 4 = 2 = 5? Never.

Thus insufficient.

IMO E

PS: If you have any kind of concerns about geometrical figure have a look at the below image for an idea. (Honestly I did not know what is a Kite (in terms of geometrical shape ))

[Rahul@gurome]

If ABCD is quadrilateral, is AB=BC=CD=DA?

1) AC is perpendicular to BD
2) AB+CD=BC+AD

Statement 1: AC is perpendicular to BD.
As there is no restrictions on the lengths of the sides, they can be same or different.

Not sufficient


Statement 2: AB + CD = BC + AD
There can be various possible combinations of lengths satisfying this condition.

Not sufficient

1 & 2 Together: Choose AB = AD and CD = AD, then statement 2 satisfies and we can make AC perpendicular to BD (as shown in the figure below) but still lengths of all the sides are not same.

Same case for AB = BC and CD = AD.

Not sufficient.

The correct answer is E.

He wants to prove that ABCD is a square with sides AB BC CD DA all equal.

No. The question is asking for whether ABCD is a rhombus or not.
Square is a special type of rhombus where all the angles are right angle.