Option 2 : 325 cm^{2}

**Given:**

BX = 15 cm

DX = 10 cm

**Concept used:**

Pythagoras theorem - In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

**Formula used:**

H^{2} = P^{2} + B^{2}

where H is hypotenuse, P is perpendicular and B is base.

**Calculation:**

In a ΔAPX

AX^{2} = AP^{2} + PX^{2} ---(1)

In a ΔBPX

BX^{2} = PX^{2} + BP^{2} ---(2)

In a ΔCQX

CX^{2} = CQ^{2} + QX^{2} ---(3)

In a ΔDRX

DX^{2} = DR^{2} + RX^{2} ---(4)

Adding equation (1) and equation (3), we get

AX^{2} + CX^{2} = AP^{2} + PX^{2} + CQ^{2} + QX^{2}

Adding equation (2) and equation (4), we get

BX^{2} + DX^{2} = PX^{2} + BP^{2} + DR^{2} + RX^{2}

∵ AP = DR, CQ = RX and QX = BP

∴ AX^{2} + CX^{2} = BX^{2} + DX^{2}

⇒ AX^{2} + CX^{2} = (15)^{2} + (10)^{2}

⇒ AX^{2} + CX^{2} = 225 + 100

⇒ AX^{2} + CX^{2} = 325

**∴ The value of (AX ^{2} + CX^{2}) is 325.**