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

upasanadatta · **Posted:** Tue Feb 07, 2012 3:01 am


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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 310 times ]

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

[Reveal] Spoiler:

56

GyanOne · **Posted:** Tue Feb 07, 2012 3:13 am

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

upasanadatta · **Posted:** Tue Feb 07, 2012 3:19 am

GyanOne wrote:

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

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...

upasanadatta · **Posted:** Tue Feb 07, 2012 3:23 am

I did like this:

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

Bunuel · **Posted:** Tue Feb 07, 2012 4:56 am

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 297 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.

VeritasPrepKarishma · **Posted:** Tue Feb 07, 2012 9:25 am

upasanadatta wrote:

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...

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

upasanadatta · **Posted:** Tue Feb 07, 2012 8:41 pm

VeritasPrepKarishma wrote:

upasanadatta wrote:

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...

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.

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

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

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