P is any point inside the rectangle ABCD. If PA = 98 cm, PB = 91 cm and PC = 21 cm, then the length of PD (in cm) is equal to:

The correct answer is 42 (Option 4).

Understanding the Property:
For any point $P$ inside a rectangle $ABCD$, British Flag Theorem applies. According to this theorem, the sum of the squares of the distances from $P$ to two opposite corners equals the sum of the squares of the distances from $P$ to the other two opposite corners.

Mathematically, this relationship is expressed as:

${PA}^2+{PC}^2={PB}^2+{PD}^2$

Step-by-Step Calculation:

Given values:

$PA=98\text{cm}$

$PB=91\text{cm}$

$PC=21\text{cm}$

Substitute the given values into the formula:

${98}^2+{21}^2={91}^2+{PD}^2$

Now, calculate the squares:

${98}^2=9604$

${21}^2=441$

${91}^2=8281$

Substitute these values back into the equation:

$9604+441=8281+{PD}^2$

$10045=8281+{PD}^2$

Isolate ${PD}^2$:

${PD}^2=10045-8281$

${PD}^2=1764$

Take the square root on both sides to find $PD$:

$PD=\sqrt{1764}$

$PD=42\text{cm}$

Thus, the length of $PD$ is equal to 42 cm.