Friday, September 18, 2009

Problem 356. Square, Interior point, Triangles, Area

Proposed Problem
Click the figure below to see the complete problem 356 about Square, Interior Point, Triangles, Area.

 Problem 356. Square, Point inside, Triangles, Area.
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Complete Problem 356
Level: High School, SAT Prep, College geometry

2 comments:

  1. we have:AB=BC=CD=AD=a
    S1+S3=(1/2*a*H1)+(1/2*a*H3)=1/2a*(H1+H3)=1/2*a*a
    S2+S4=(1/2*a*H2)+(1/2*a*H4)=1/2a*(H2+H4)=1/2*a*a
    then:S1+S3=S2+S4=1/2*a*a=1/2s.

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  2. Let l be the side of the square ABCD and h1, h2, h3, h4 be respectively the heights of triangles PAB, PBC, PCD, PDA. We have:
    S↓1=lh↓1/2, S↓2 = lh↓2/2, S↓3=l h↓3/2, S↓4=l h↓4/2. Therefore:

    S↓1+ S↓3=l(h↓1+ h↓3)/2= l↑2./2,
    S↓2+ S↓4=l(h↓2+ h↓4)/2= l↑2./2.
    QED, Ianuarius.

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