# find length of PB

#### Albert

##### Well-known member
please find the length of PB

#### caffeinemachine

##### Well-known member
MHB Math Scholar
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 $O$.

#### Opalg

##### MHB Oldtimer
Staff member

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.

#### Albert

##### Well-known member
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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