British Flag Theorem

This theorem is famous for Math Olympians but never heard inside our classroom. I’m happy to share this to my readers for you to know this very interesting theorem.

Given a rectangle ABCD, we choose point P inside the rectangle and the distance from point A to P is a, B to P is b, C to P is c and D to P is d.

bgtfinal

Then,

a^2+c^2=b^2+d^2

Sample Problem:

1. Let ABCD be a rectangle and let P be a point inside the rectangle. If PA = 8, PB=4 and PD = 7, then what is the length of PC?

Solution:

PA=a, PC=c, PB=b, PD=d

 Using the formula,

a^2+c^2=b^2+d^2

8^2+c^2=4^2+7^2

c^2=4^2+7^2-8^2

c^2=16+49-64

c^2=1

c=1

Therefore, PC=1.

2. ABCD is a rectangle and P is inside the rectangle. AP=x-1, BP=x+3, PC=x+2, PD=x-4. How long is BP?

Solution:

PA=a, PC=c, PB=b, PD=d

 Using the formula,

a^2+c^2=b^2+d^2

(x-1)2+( x+2)2=( x+3)2+( x-4)2

Simplify and solve for x we have,

x=5

BP=x+2

=5+2

BP=7

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