Common Mistakes in Geometry Questions - Exercise Question #1

Question

Exercise Question #1     Common Mistakes in Geometry Questions

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ABCD is a quadrilateral as shown in the figure above. Find the area of the quadrilateral.

(1) The diagonals bisect each other at  90^o  and they are equal. 
(2) The length of each diagonal is 10 cm.

Answer Choices

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

Next Question

Detailed solution will be posted soon.

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EgmatQuantExpert · Answer

Note:    This questions is related to the article on  Common Errors in Geometry

Kindly go through the article once, before solving the question or going through the solution.

Official Solution

Given:   ABCD is a quadrilateral.

Analysing statement 1:

The first statements states :

The diagonals bisect each other at  90^o  and they are equal.

From the above statement, we can conclude that  the quadrilateral is a square.

But to find the area of the quadrilateral we need the  length of each side, which is not given .

Hence statement 1 is   not sufficient   to answer the question.

Analysing statement 2 :

The length of each diagonal is 10 cm.

Using only the 2nd statement, we cannot find the area of the quadrilateral, since we don't know what kind of quadrilateral it is.

Hence statement 2 is   not sufficient   to answer the question.

Combining statement 1 and 2:

Using both the statements, we can conclude that the quadrilateral is a square and the length of it's diagonal is 10cm.

Therefore, AB = BC = CD = AD

AC = BD = 10 (diagonals)

https://e-gmat.com/blogs/wp-content/uploads/2016/11/Screen-Shot-2016-11-22-at-4.52.43-PM.png

Since ABCD is a square angle ABC = 9 0^o , therefore we can infer that the diagonal(AC) is the hypotenuse and the sides AB and BC are the perpendicular and base.

Hence we can write A B^2  + B C^2  = A C^2

2A B^2  = 1 0^2 .............(i)

And we know the area of the square = (side )^2  = A B^2

We can find the value of AB from (i) and hence we can find the area of the square!

Hence   combining both the statements we can find out the answer.

Correct Option : C

Thanks,
Saquib
Quant Expert
e-GMAT

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ccooley · Answer

An interesting variant is to figure out what the answer should be if the word 'bisect' is removed from statement (1)

EgmatQuantExpert · Answer

Hey Everyone, The official solution has been posted. Kindly go through it and if you have any doubts feel free to post your query.

Thanks, Saquib Quant Expert e-GMAT

TheKingInTheNorth · Answer

Could you please share the relation between all the polygon .
For Eg . How can a square be a Rhombus .

utkarshthapak · Answer

Hi  abhisheknandy08 ,

There can be many types of polygons based on number of sides. 
Now, there are regular polygons and non-regular polygons.
Regular polygons have all sides equal but non-regular polygons can have any shape.

For the purpose of understanding this question, I'll stick with regular and non regular 4 sided polygon which have opposites sides parellel.

So, any polygon which has 2 of its opposite sides parallel and equal will be a parallelogram   (so rectangle, rhombus and square are all parallelogram)  
If in a parallelogram we make 2 sides equal (not all 4) and all angles 90 degree, we get a rectangle.
If we make all sides equal but the angles are not equal to 90 degree, we get a rhombus 
Now if we make all angles 90 degree and all sides equal, we get a square   (all squares are rhombus with 90 degree angles or all squares are rectangle with equal sides and all rhombus are parallelogram)  .

Hope it helps.

GMATBusters · Answer

in that case, Answer would have been E.

kckartick · Answer

Dear Experts,

Statement 1 says that the diagonals bisect each other at 90 degrees and that they are equal. This is could mean that the given quadrilateral could just be a rectangle instead of a square. If this is the case, then combining statement 1 and 2, we still cannot find the area since two adjacent sides of a rectangle can be different.

Please clarify.

EgmatQuantExpert · Answer

Hey  kckartick ,

Not in each and every rectangle the diagonals bisect each other at 90 degrees. It happens only when the rectangle is a square - in that case, the lengths of the adjacent sides remain same.

Helium · Answer

ccooley  -

Answer - C

Diagonals are still equal and perpendicular - So Square.

venkivety · Answer

from st-1 we can say

Diagonals are equal and perpendicular - So Square. bt no data on the side/diagonal measurements.

st-2 only gv value of the diagonal bt we can't infer the type of quadrilateral. NS

1&2 square and a value of the diagonal. we can find the area.
Answer - C

avi94 · Answer

Just confirming - A rhombus also has diagonals that are perpendicular bisectors right? in that case how can we assume it is a square and not a rhombus?

VeritasPrepErika · Answer

Quick recommendation: draw this out. Try drawing a particularly dramatic rhombus (extreme angle measurements).

You should find that while the diagonals do intersect each other at 90 degree angles,  they aren't the same length , as required by the second half of Statement 1. Only a square or a rectangle will give us diagonals of equal length — any type of parallelogram (rhombus included) with angle measurements other than 90 degrees will have diagonals with different lengths (the diagonal connecting the two larger angles will be shorter than the diagonal connecting the two smaller angles).

So the first half of Statement 1 ensures that all of the sides are the same length (leaving only squares and rhombuses), and the second half of Statement 1 ensures that the all of the angles are the same measurement (leaving only rectangles and squares). A square is the only option that fulfills both requirements.

Make sure to not to forget about pieces of the question! It's easy to do and will almost always lose us the question.

philipssonicare · Answer

Great explanation for why statement 2 is insufficient https://www.quora.com/How-do-I-calculat ... l-is-given

Sarkar93 · Answer

Hi, 
In statement 2, it is stated that the length of each diagonal is 10 cm. This implies that both the diagonals are equal in length. I was under the impression that square is the only quadrilateral that has both diagonals equal in length. Or is there any other quadrilateral that has it's diagonals equal ?

saurabh9gupta · Answer

EgmatQuantExpert

Statement 1 can also mean that the quadrilateral is a Square or a Rhombus since it is the property of both Square and Rhombus to have diagonals that are perpendicular bisectors.

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Common Mistakes in Geometry Questions - Exercise Question #1

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