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Area of shaded region_PS

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Area of shaded region_PS [#permalink]

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New post 28 Nov 2009, 04:18
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A square of side 2 cms (1+1) is given. Find the area of the shaded region?
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Re: Area of shaded region_PS [#permalink]

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New post 28 Nov 2009, 06:33
The area of each of the colored triangle is 1/2*1/2*1 =1/4

There are four such triangles so total area of the colored region is 1
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Re: Area of shaded region_PS [#permalink]

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New post 28 Nov 2009, 11:14
nope buddy,
how can you prove that base of triangle is 1/2 ???
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Re: Area of shaded region_PS [#permalink]

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New post 28 Nov 2009, 14:44
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Consider Right tri'gle HFC. We have to figure out if R is mid-point of OG.

Two right tri'gles HFC and HOR are similar triangles by angle-side-angle (asa) property.
1) Angle RHO is same as Angle CHF.
2) HO and HF are in straight line, given that O is mid-point of HF, HO/HF = 1/2.
3) Angle HOR is same as Angle HFC = 90 degree.

So HO/HF = HR / HC = OR / FC = 1/2.
Hence, OR is half of FC. And OG = FC = 1. Therefore, R is mid-point of OG.

You can take it from here.
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Re: Area of shaded region_PS [#permalink]

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New post 28 Nov 2009, 19:59
brothers wrote:
nope buddy,
how can you prove that base of triangle is 1/2 ???


And also, as Square HOGD and Square OGSF are congruent and occupy exactly half the area of the rectangle HDFC. The diagonal HC cuts OG exactly at the mid point.
Re: Area of shaded region_PS   [#permalink] 28 Nov 2009, 19:59
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