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Question
ABCD is a trapezium with parallel sides AB =a and DC = b. If E and F are mid-points of non-parallel sides AD and BC respectively, then the ratio of areas of quadrilaterals ABFEand EFCD is
Options
a : b
(a + 3b): (3a + b)
(3a + b) : (a + 3b)
(2a + b) : (3a + b)
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Solution
Given: (1) ABCD is a trapezium, with parallel sides AB and DC
(2) AB = a cm
(3) DC = b cm
(4) E is the midpoint of non parallel sides AD.
(5) G is the midpoint of non parallel sides BC.
To find: Ratio of the area of the Quadrilaterals ABFE and EFCD.
Calculation: We know that, ‘Area of a trapezium is half the product of its height and the sum of the parallel sides.’
Since, E and F are mid points of AD and BC respectively, so h1 = h2
Area of trapezium ABFE
Area of trapezium ABFE`=1/2 (a + x) h_1 = 1/2 (a + x)h`
Area of trapezium EFCD = `1/2 (b+x)h_2 = 1/2 (b+ x) h`
Area of trapezium ABCD `= 1/2 (a+ b) (h_1 + h_2 ) = (a+b) h`
Now, Area (trap ABCD) = area (trap EFCD) + Area (ABFE)
Therefore,
(a + b) h = `1/2 ( a + x) h + 1/2 (b+x) h`
`⇒ A +b = (a+b)/2 +x`
`⇒x = (a+b)/2`
Thus,
`(Area (ABFE))/(Area (EFCD)) = (a+x)/(b +x) = (a +(a+b)/2)/(b + (a+b)/2)`
`⇒ (Area (ABFE))/(Area (EFCD)) = (3a +b)/(a + 3b)`
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