The above shows a rectangle with as a point in the interior of this rectangle.
We are given the distances .
Find the length of .
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Let D E = x , E P = y , D C = p , and A D = l .
△ D P E → D E 2 + E P 2 = x 2 + y 2 = 9
△ A P F → P F 2 + A F 2 x 2 + ( l − y ) 2 x 2 + y 2 + l 2 − 2 l y l 2 − 2 l y + 9 l 2 − 2 l y = 2 5 = 2 5 = 2 5 = 2 5 = 1 6
△ B P H → P H 2 + H B 2 ( p − x ) 2 + l 2 − 2 l y + y 2 ( p − x ) 2 + y 2 + 1 6 ( p − x ) 2 + y 2 = ( p − x ) 2 + ( l − y ) 2 = 1 6 0 = 1 6 0 = 1 6 0 = 1 4 4
△ P C H → C P 2 C P 2 C P 2 = P H 2 + C H 2 = ( p − x ) 2 + y 2 = 1 4 4 → C P = 1 2
Alternative
By British Flag Theorem, A P 2 + C P 2 = B P 2 + D P 2
2 5 + C P 2 = 1 6 0 + 9
C P 2 = 1 4 4 → C P = 1 2 .