Proposed Problem

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

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Complete Problem 356

Level: High School, SAT Prep, College geometry

## Friday, September 18, 2009

### Problem 356. Square, Interior point, Triangles, Area

Labels:
area,
interior point,
square,
triangle

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we have:AB=BC=CD=AD=a

ReplyDeleteS1+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.

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:

ReplyDeleteS↓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.