Calculate the area of quadrilateral ABCD, in which ∠ABD = 90°, triangle BCD is an equilateral triangle of side 24 cm and AD = 26 cm.

For Δ ABD,

By using the Pythagoras theorem,

Base² + Height² = Hypotenuse²

⇒ AB² + BD² = AD²

⇒ AB² + (24)² = (26)²

⇒ AB² + 576 = 676

⇒ AB² = 676 - 576

⇒ AB² = 100

⇒ AB = $\sqrt{100}$

⇒ AB = 10 cm

Area of Δ ABD = $\dfrac{1}{2}$ x AB x BD

= $\dfrac{1}{2}$ x 10 x 24 cm²

= 5 x 24 cm²

= 120 cm²

Area of equilateral triangle BCD = $\dfrac{\sqrt{3}}{4}$ x side²

= $\dfrac{\sqrt{3}}{4}$ x 24²

= $\dfrac{\sqrt{3}}{4}$ x 576

= 144 ${\sqrt{3}}$ cm²

= 249.41 cm²

Total area of quadrilateral ABCD = Δ ABD + Δ BCD

= 120 + 249.41 cm²

= 369.41 cm²

Hence, the area of quadrilateral ABCD is 369.41 cm².