Question

If a quadrilateral with sides length a, b, c, d can be inscribed in one circle and circum scribed about another circle then radius of later circle is –

• A. $\frac{ac + bd}{2(a + b + c + d)}$
• B. $\frac{\sqrt{a^{4} + b^{4} + c^{4} + d^{4} - abcd}}{2(a + b + c + d)}$
• C. $\frac{2\sqrt{abcd}}{a + b + c + d}$
• D. None of these

Answer 1

Answer: (C)

$\frac{2\sqrt{abcd}}{a + b + c + d}$

Answer 2

Area of quadrilateral = $\frac { 1 } { 2 }$(ar + rb + rc + rd)

= $\sqrt { \mathrm { abcd } }$=

⇒ r = $\frac { 2 \sqrt { a b c d } } { a + b + c + d }$

[Q Area = $\sqrt { ( s - a ) ( s - b ) ( s - c ) ( s - d ) }$ =

In this case

s – a = c

s – b = d

s – c = a

s – d = b