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In the figure, ABCD is a rectangle and points E, F, G and H
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12 Apr 2019, 03:23
12 Apr 2019, 03:23
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In the figure, ABCD is a rectangle and points E, F, G and H are midpoints of its sides. What is the ratio of the area of the shaded region to the area of the un-shaded region in the rectangle?
(A) 1 : 1
(B) 1 : 2
(C) 2 : 1
(D) 1 : 3
(E) 3 : 1
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Re: In the figure, ABCD is a rectangle and points E, F, G and H
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07 Jun 2020, 03:25
07 Jun 2020, 03:25
Let side AB = a and side AD = b
If we find the area of the shaded region then Area of Un-shaded region = Area of ABCD - area of the shaded region
So, lets find the area of the shaded region first.
Area of Shaded region = Area of the four triangles. Note that all 4 shaded triangles are congruent (Same dimensions)
So, Area of Shaded region = 4 * Area of one shaded triangle = 4 * Area of AEH
Let's find out area of AEH
AE = \(\frac{1}{2}*AB\) = \(\frac{a}{2}\) and AH = \(\frac{b}{2}\)
Area AEH = \(\frac{1}{2}*AE*AH\) = \(\frac{1}{2}*AE*AH\) = \(\frac{1}{2}*\frac{a}{2}*\frac{b}{2}\) = \(\frac{a*b}{8}\)
=> Area of Shaded region = 4 * Area of AEH = \(4*\frac{a*b}{8}\) = \(\frac{a*b}{2}\)
Area of Un-shaded region = Area of ABCD - area of the shaded region = a*b - \(\frac{a*b}{2}\) = \(\frac{a*b}{2}\)
=> Area of Un-shaded region = area of the shaded region
So, ratio of the area of the shaded region to the area of the un-shaded region in the rectangle = 1:1
So, answer will be A
Hope it helps!
If we find the area of the shaded region then Area of Un-shaded region = Area of ABCD - area of the shaded region
So, lets find the area of the shaded region first.
Area of Shaded region = Area of the four triangles. Note that all 4 shaded triangles are congruent (Same dimensions)
So, Area of Shaded region = 4 * Area of one shaded triangle = 4 * Area of AEH
Let's find out area of AEH
AE = \(\frac{1}{2}*AB\) = \(\frac{a}{2}\) and AH = \(\frac{b}{2}\)
Area AEH = \(\frac{1}{2}*AE*AH\) = \(\frac{1}{2}*AE*AH\) = \(\frac{1}{2}*\frac{a}{2}*\frac{b}{2}\) = \(\frac{a*b}{8}\)
=> Area of Shaded region = 4 * Area of AEH = \(4*\frac{a*b}{8}\) = \(\frac{a*b}{2}\)
Area of Un-shaded region = Area of ABCD - area of the shaded region = a*b - \(\frac{a*b}{2}\) = \(\frac{a*b}{2}\)
=> Area of Un-shaded region = area of the shaded region
So, ratio of the area of the shaded region to the area of the un-shaded region in the rectangle = 1:1
So, answer will be A
Hope it helps!
gmatclubot
Re: In the figure, ABCD is a rectangle and points E, F, G and H [#permalink]
07 Jun 2020, 03:25
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