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# A quadrilateral has a perimeter of 16. Which of the followin

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A quadrilateral has a perimeter of 16. Which of the followin [#permalink]  29 Oct 2017, 02:05
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4% (00:36) correct 95% (01:15) wrong based on 21 sessions
A quadrilateral has a perimeter of 16. Which of the following alone would provide sufficient information to determine the area of the quadrilateral. Choose ALL that apply.

[A] The quadrilateral contains equal sides

[B] The quadrilateral is formed by combining two isosceles right triangles

[C] Two pairs of congruent angles are in a 2:1 ratio

[D] The width is 4o% of the length and all angles are of equal measure

[E] If the perimeter was decreased by 50%, the area would decrease by 25%

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[Reveal] Spoiler: OA
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Re: A quadrilateral has a perimeter of 16. Which of the followin [#permalink]  25 Apr 2018, 03:15
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Can someone provide the explanation for this please?
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Re: A quadrilateral has a perimeter of 16. Which of the followin [#permalink]  27 Apr 2018, 00:16
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Let us start with $$A$$
The quadrilateral has equal sides so general perception could be that this is a square however there is one more quadrilateral that has all four sides equal to one another but has an area different from square i.e Rhombus. area of square = side$$^2$$ and area of rhombus = $$1/2 D1 * D2$$ so a is not true
$$B$$
The quadrilateral contains two right angled triangle hence there are two $$45-45-90$$ triangle as such all four sides of the quadrilateral should be equal but this does not necessarily mean the quadrilateral is a square as it could again be a rhombus

$$C$$
In this case we can be certain that the quadrilateral is not a square as all the angles of square are $$90$$ degrees. This could be a parallelogram however there could be multiple scenarios for a perimeter of $$16$$ for eg. a side of $$6$$ and $$2$$ or $$7$$ and $$1$$ or $$5$$ and $$3$$ with a $$2:1$$ ratio of ($$60,120$$) degrees
In all of these variation in sides area will be different also notice that we need a height to get the area of a parellelogram which is not given nor can be derived hence c is also false

$$D$$
For a quadrialteral all angles add to $$360$$ degrees hence if all angles are equal $$4x =360$$and as such each angle $$(x)$$ = $$90$$ degrees furthermore given that width is $$0.4$$ of length the quadrilateral is not a square or rhombus but a rectangle
we can find the exact length and width of the rectangle $$2(l + 0.4l) = 16$$
or $$1.4 l = 8$$
therefore $$l = 5.70$$ approx
similarly we can get the width $$= 2.28$$
and area = $$12.996$$
hence $$D$$ is true

$$E$$
Let us take a rectangle with sides $$6$$and$$2$$
Initial perimeter = $$16$$
50% decrease in perimeter so new perimeter = $$8$$
This should bring about $$25$$% reduction in area
original are
a = $$6*2 =12$$
new area = $$12*0.75 = 9$$

However a combination of length and breadth with perimeter $$16$$ for a rectangle can be many such as $$7,1$$; $$5,3$$
In each case the area is not same and hence when the perimeter and area is reduced by 50% and 25% respectively we get a different area for different set of sides

for $$7,1$$ original area = $$7$$ after reduction = $$5.25$$
for $$5,3$$ original area = $$15$$ after reduction = $$11.25$$
We cannot be sure of the area hence E is also false
only D
_________________

This is my response to the question and may be incorrect. Feel free to rectify any mistakes

Re: A quadrilateral has a perimeter of 16. Which of the followin   [#permalink] 27 Apr 2018, 00:16
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