Proposed Problem

Click the figure below to see the complete problem 357 about Square, Exterior point, Triangles, Area.

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Complete Problem 357

Level: High School, SAT Prep, College geometry

## Friday, September 18, 2009

### Problem 357. Square, Exterior point, Triangles, Area

Labels:
area,
exterior point,
square,
triangle

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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↓4- S↓2=l(h↓4- h↓2)/2= l↑2./2.

QED, Ianuarius.

Draw EPF//BC, E on AB extended and F on DC extended

ReplyDeleteLet EP = x and EB = y

So S1 = xa/2, S3 = (a-x)a/2,

S2 = ya/2 and S4 = (a+y)a/2

So S1+S3 = S4-S2= a^2/2 = S/2

Sumith Peiris

Moratuwa

Sri Lanka