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Common Mistakes in Geometry Questions - Exercise Question #1

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Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post Updated on: 07 Aug 2018, 04:09
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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.


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Detailed solution will be posted soon.


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Originally posted by EgmatQuantExpert on 22 Nov 2016, 05:32.
Last edited by EgmatQuantExpert on 07 Aug 2018, 04:09, edited 5 times in total.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post Updated on: 01 Dec 2016, 01:47
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)

Image

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



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Saquib
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Originally posted by EgmatQuantExpert on 22 Nov 2016, 11:13.
Last edited by EgmatQuantExpert on 01 Dec 2016, 01:47, edited 2 times in total.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 22 Nov 2016, 13:15
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An interesting variant is to figure out what the answer should be if the word 'bisect' is removed from statement (1) :)
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 27 Nov 2016, 09:23
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 20 Jun 2017, 17:16
Could you please share the relation between all the polygon .
For Eg . How can a square be a Rhombus .
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 21 Jun 2017, 11:04
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abhisheknandy08 wrote:
Could you please share the relation between all the polygon .
For Eg . How can a square be a Rhombus .


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.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 14 Aug 2018, 22:30
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.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 14 Aug 2018, 22:58
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kckartick wrote:
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.


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.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 24 Aug 2018, 20:34
ccooley wrote:
An interesting variant is to figure out what the answer should be if the word 'bisect' is removed from statement (1) :)


ccooley -

Answer - C

Diagonals are still equal and perpendicular - So Square.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 17 Jun 2019, 02:31
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
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 27 Jun 2019, 07:50
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?
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Re: Common Mistakes in Geometry Questions - Exercise Question #1  [#permalink]

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New post 27 Jun 2019, 08:16
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avi94 wrote:
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?

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.
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Re: Common Mistakes in Geometry Questions - Exercise Question #1   [#permalink] 27 Jun 2019, 08:16
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