Geometry MGMAT question bank #6

[Gmat2Go]

Is quadrilateral ABCD a rectangle?
(1) Line segments AC and BD bisect one another.

(2) Angle ABC is a right angle.


OA is C.

From 2) I can see that its either square or rectangle so cross off b and d
From 1) I can still only see its a square

I can't tell why C tells me its rectangle but not a square? When the lines bisect each other, doesn't it create 45 degree angles, thus 45-45-90, making all sides x radical 2? THus, its a square? Please advise!

[givemeanid]

I can't tell why C tells me its rectangle but not a square? When the lines bisect each other, doesn't it create 45 degree angles, thus 45-45-90, making all sides x radical 2? THus, its a square? Please advise!

When diagonals bisect, they are not necessarily perpendicular bisectors of each other. For a rectangle to be a square, the diagonals have to be perpendicular bisectors, not just bisectors.

[Guest]

So just to follow up.

Is the Answer C or E?

If C, givemeanid, or someone else - can you please explain in further detail?

Thanks!

[givemeanid]

So just to follow up.

Is the Answer C or E?

If C, givemeanid, or someone else - can you please explain in further detail?

Thanks!

1. When diagonals bisect each other, the quadrilateral is a parallelogram. However, this is not enough to say that the quad is a rectangle.
INSUFFICIENT.

2. Angle ABC = 90 degrees does not mean the quad is rectangle. We do not know anything else.
INSUFFICIENT.

Now, combining together, we know it is a parallelogram with one angle = 90 degrees. This is sufficient to say that the quad is a rectangle. (To figure this, since one angle is 90, the opposite angle is 90 too. In a parallelogram, opposite angles are equal. So, we have a pair of angles, each = 90. Since the sum of all 4 angles = 360, sum of the other pair = 360 - 90*2 = 180. Since, these angle are also equal to each other, they are also 90 degrees each. So, all angles are 90 degrees and diagonals bisect. The quad is a rectangle.)
SUFFICIENT.

Answer is C.

[Gmat2Go]

Is it true that all squares are also rectangles?

[givemeanid]

Is it true that all squares are also rectangles?

Yes. All squares are rectangles. However, the inverse is not true.

[StaceyKoprince]

Nice, givemeanid. Keep up the great work!

[tlc]

With the given two statements, couldn't the quadrilateral also be a rhombus? Definition of a rhombus: All sides are equal and opposite angles are equal.

Therefore,

(1) A rhombus can have line segments AC and BD bisect one another
(2) A rhombus can also have one right angle

Please explain why the two statements don't also suggest that the quadrilateral could be a rhombus.

Thanks-

[tlc]

Is anyone monitoring this post? Thanks-

[JonathanSchneider]

It COULD be a rhombus, but it doesn't have to be. A rhombus must have four equal side lengths. This is not known to be the case here. Furthermore, a rhombus does not have to have a right angle at its side (though it could - this would be called a square, which is an example of a rhombus). As a result, we only know that the shape is a rectangle.

[ap11]

Why is Answer not E?

Are bisectors always perpendicular? See statement 1 of the problem.

I think the answer should be E because, with the given information, it could also be a trapezoid for statement 1 or statement 2. Please help!

And another quick question: How are we supposed to know when to apply the rules of squares vs rhombus vs rectangle?

[RonPurewal]

Are bisectors always perpendicular?

draw a random rectangle (not a square).

the diagonals bisect each other (= divide each other into equal halves).
so, they are bisectors.
are they perpendicular?

[RonPurewal]

it could also be a trapezoid

nope. in a trapezoid the diagonals don't, and can't, bisect each other.

[RonPurewal]

And another quick question: How are we supposed to know when to apply the rules of squares vs rhombus vs rectangle?

i don't know which 'rules' you're talking about, but, personally, i would just draw diagrams and look at them with my eyes.

[KarthikO685]

Can't a Kite figure satisfy both statement 1 and 2 ?