MHB What Does PB Refer to in Measurements?

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The discussion focuses on determining the length of PB in a geometric context involving a square and the rotation of points. By rotating the square, the position of point P becomes collinear with points P and O, leading to the conclusion that triangles APB and BP'C are congruent. Given the lengths of segments AP and BP as 5x and 14x, respectively, the Pythagorean theorem is applied to find the length of AB, resulting in an area calculation for the square. Ultimately, it is established that PB measures 42 cm. The geometric relationships and calculations confirm the final measurement of PB.
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please find the length of PB

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Albert said:
please find the length of PB

View attachment 1178
HINT:

Rotating the square by 90 degrees in the clockwise direction, the new position of $P$ is collinear with $P$ and $O$.
 
http://mathhelpboards.com/attachments/challenge-questions-puzzles-28/1178d1376578218-find-length-pb-length-pb.jpg
caffeinemachine said:
HINT:

Rotating the square by 90 degrees in the clockwise direction, the new position of $P$ is collinear with $P$ and $O$.
Good idea! But what I would do is to rotate the square by 90 degrees in an anticlockwise direction. The new position (call it $P'$) of $P$ will be on the line $PB$. The lines $PB$, $P'C$ will be perpendicular, and the triangles $APB$, $BP'C$ will be congruent. Therefore $\angle APB$ is a right angle. If the lengths of $AP$, $BP$ are $5x$ and $14x$ then by Pythagoras $AB$ will be $\sqrt{221}x$ and the area of the square is $221x^2$. Since $1989 = 9\times 221$ it follows that $x=3$ and so $PB = 42$cm.
 
It is easy to see that four points A,B,O,P are
located on the same circle
let AP=5x ,BP=14X
$AP^2+BP^2=221x^2=AB^2=1989$
$\therefore x=3cm$
$BP=14x=42cm$
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