SOLUTION: ABCD is a trapezoid. E is a point between B and C. 2|BE| = |EC|, 5|AB| = 3|DC|
Area of DEC is 10. So what is area of AED?
Algebra ->
Surface-area
-> SOLUTION: ABCD is a trapezoid. E is a point between B and C. 2|BE| = |EC|, 5|AB| = 3|DC|
Area of DEC is 10. So what is area of AED?
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See the steps below to see how I got that answer
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A diagram isn't 100% mandatory, but for me it really helps.
Based on the diagram we have...
x = vertical distance from point C to point E
y = vertical distance from point E to point B
z = |AB| = distance from A to B
w = |DC| = distance from C to D
T1 = area of the gray triangle ABE
T2 = area of the green triangle AED
T3 = area of the red triangle DEC
Because 2|BE| = |EC| and because E is on segment BC, this means that the vertical component from C to E is twice that as the vertical component from E to B.
In other words, x = 2y
Also,
5|AB| = 3|DC|
5*z = 3*w
w = 5z/3
We will use the two equations x = 2y and w = 5z/3 for the sections below
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T3 = area of triangle DEC (red triangle)
T3 = (base*height)/2
T3 = x*w/2
10 = x*w/2 .... plug in the given area for triangle DEC (which was 10 square units)
2*10 = x*w
20 = x*w
20 = (2y)*(5z/3) .... plug in x = 2y; plug in w = 5z/3
20 = 10yz/3
3*20 = 10yz
60 = 10yz
6 = yz
yz = 6 ... this will be used later
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A = area of trapezoid
A = height*(base1+base2)/2
A = h*(b1+b2)/2
A = (x+y)*(w+z)/2
A = (2y+y)*(w+z)/2 .... plug in x = 2y
A = 3y*(w+z)/2
A = 3y*(5z/3+z)/2 .... plug in w = 5z/3
A = 3y*(5z/3+3z/3)/2
A = 3y*(8z/3)/2
A = 3y*8z/6
A = 3y*4z/3
A = 12yz/3
A = 4yz
A = 4*6 .... replace yz with 6 (since it was found earlier that yz = 6)
A = 24
The area of the trapezoid is 24 square units
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T1 = area of gray triangle ABE
T1 = (base*height)/2
T1 = z*y/2
T1 = yz/2
T1 = 6/2 .... replace yz with 6
T1 = 3
The gray triangle T1 has area 3 square units
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T2 = area of green triangle AED (what we want)
T2 = A - T1 - T3
T2 = 24 - 3 - 10
T2 = 11
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