ABCDis a rhombus (see figure). ABE is a right triangle. AB

Question

ABCD is a rhombus (see figure). ABE is a right triangle. AB is 10 meters. The ratio of the length of CE to the length of EB is 2 to 3. What is the area of trapezoid AECD?

Attachment:

Rhombus.PNG [ 10.76 KiB | Viewed 11505 times ]

IMO the area is 76,,,,,the answer that is given is

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56

Bunuel · Accepted Answer

ABCD is a rhombus (see figure). ABE is a right triangle. AB is 10 meters. The ratio of the length of CE to the length of EB is 2 to 3. What is the area of trapezoid AECD?

Attachment:

Rhombus.PNG [ 10.76 KiB | Viewed 11385 times ]

Since ABCD is a rhombus all its four sides are equal in length: AB=AD=CB=DC=10;

Next, as the ratio of CE/EB=2/3, then CE+EB=2x+3x=CB=10 --> 5x=10 --> x=2 --> CE=4 and EB=6. From right triangle ABE: EB^2+AE^2=AB^2 --> 6^2+AE^2=10^2 --> AE=8;

Area of trapezoid \(are=a*\frac{b_1+b_2}{2}\), where b1, b2 are the lengths of the two bases a is the altitude of the trapezoid. Hence, the are of trapezoid AECD is \(area=AE*\frac{AD+CE}{2}=8*\frac{10+4}{2}=56\).

Hope it's clear.

GyanOne · Answer

Area of the trapezoid = (1/2) * (CE+AD) * (AE)

If AB = 10, then AE=8 and EB=6 (AE=6 and EB=8 is not possible since EB must be divisible by 3) 
=> CE = 4
=> AD = 10

So area = (1/2) * (14) * 8
= 56

Dreaming · Answer

But AE is not the altitude for trapezoidAECD is it? it is only one of the sides...........altitude will be a line from E dropped on AD...

Dreaming · Answer

I did like this:

areea of trapezium=area of rhombus ABCD-triangle AEB
 =10x10-1/2x6x8
 =76

KarishmaB · Answer

Notice that AE is perpendicular to BC so it is also perpendicular to CE i.e. the side of the trapezoid. AE is perpendicular to AD as well since AD is parallel to CE. 
So AE is the perpendicular distance between AD and CE i.e. between the parallel sides of the trapezium. Therefore, it is the required altitude. (Hide the ABE triangle with your hand and twist your neck to the left to see the shape of AECD)

Since CE:EB = 2:3, CE = 4 and EB = 6 (Since CB = 10)
Therefore, AE = 8 (Using pythagorean theorem)

Area of the trapezoid AECD = (1/2)*(4+10)*8 = 56

Method 2:
You can also do it by subtracting the area of triangle ABE from rhombus ABCD
Area of rhombus ABCD = Altitude*Base = 8*10 = 80
Area of triangle = (1/2)*6*8 = 24
Area of rhombus = 80 - 24 = 56

Dreaming · Answer

Thank you so much Karishma! I feel like a complete idiot after your explanation :oops:

How could i be soo blind....anyway thanks for clearing up everything

Nunuboy1994 · Answer

To cover up the confusion- this shape simply represent a right trapezoid- the figure is nearly an optical illusion- essentially the GMAT is making sure you are using mathematical principles in order to derive results rather than immediately assumptions about symmetry and proportions of a figure from its appearance alone. The side that is dotted is actually a straight line, if you turn your head to the left- or alternatively notice that is perpendicular to the two other sides of the rhombus; therefore we need only the length of the dotted line and the two other parallel sides of the rhombus. Because we are given a proportion, CE to EB must be 2 to 3, we can use algebra to solve. How? We know that the figure is a rhombus- a rhombus is an equilateral parallelogram. If we know one side length of a rhombus then we know the side lengths of ALL the rest of the sides. (note: unlike a square, we cannot divide the area of a rhombus by its number of sides in order to calculate the lengths of the rhombus).

2(x) + 3(y) =10
2(2) + 3(2) = 10 
 4 + 6 = 10

CE= 4 
EB = 6

Area of a Rhombus = 8* (10+4) /2

Nunuboy1994 · Answer

Here is a link https://mathworld.wolfram.com/RightTrapezoid.html

bumpbot · Answer

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ABCDis a rhombus (see figure). ABE is a right triangle. AB

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