$2$ . GEOMETRY: Let $A B C D$ be a rectangle and $P$ be an inner point such that $P A=5$ and $P C=13$ . $P$ is not directly on the rectangle. What are the possible values of $P B+P D$ ? The diagram below is not drawn to scale.

Full solution

2

. GEOMETRY: Let

A B C D

be a rectangle and

P

be an inner point such that

P A=5

and

P C=13

P

is not directly on the rectangle. What are the possible values of

P B+P D

? The diagram below is not drawn to scale.

Apply British Flag Theorem: Use the British Flag Theorem which states that for any point $P$ inside a rectangle, the sum of the squares of the diagonals ( $PA$ and $PC$ ) is equal to the sum of the squares of the other two distances ( $PB$ and $PD$ ). $\newline$ So, $PA^2 + PC^2 = PB^2 + PD^2$ .
Calculate $PA^2$ and $PC^2$ : Calculate $PA^2$ and $PC^2$ . $\newline$ $PA^2 = 5^2 = 25$ . $\newline$ $PC^2 = 13^2 = 169$ .
Find sum of squares: Add $PA^2$ and $PC^2$ to find the sum of the squares of $PB$ and $PD$ . $\newline$ $25 + 169 = 194$ . $\newline$ So, $PB^2 + PD^2 = 194$ .
Determine minimum value: Since $PB$ and $PD$ are lengths, they must be positive. The minimum value for $PB+PD$ is when $PB=PD$ , because of the triangle inequality. $\newline$ Let's assume $PB=PD$ for the minimum value.
Solve for $PB=PD$ : If $PB=PD$ , then $2 \times PB^2 = 194$ . $\newline$ Divide both sides by $2$ to find $PB^2$ . $\newline$ $PB^2 = \frac{194}{2} = 97$ .
Identify calculation error: Find the value of $PB$ by taking the square root of $PB^2$ . $PB = \sqrt{97}$ .But wait, we made a mistake here. We can't find the exact value of $PB$ because we don't know the lengths of the sides of the rectangle. We only know that $PB+PD$ must be greater than or equal to $2\sqrt{97}$ .