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%


Kudos for correct solution.

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