Geometry Practice Test 3

Let the center of circle C. AB and CD two chords of Length 2a, 2b interest at point P. Let PB = X and PC = y units
Draw Perpendicular OM ⊥ AB
ON ⊥ CD
Join OP = C
Perpendicular from the center bisects the chord
AM = BM = a units
CN = ND = b units
⇒ PM = a - x
PN = b - y

As OMPN is a rectangle and ON = PM = a - x
PN = OM = b - y
From the rectangle OMPN, diagonal
${\mathit{c}}^2$ = ${\left(\mathit{a}-\mathit{x}\right)}^2+{\left(\mathit{b}-\mathit{y}\right)}^2$ ......(i)
From the right triangle OAM,
$\mathit{O}{\mathit{A}}^2$ = $\mathit{A}{\mathit{M}}^2+\mathit{O}{\mathit{M}}^2$
${\mathit{r}}^2$ = ${\mathit{a}}^2+{\left(\mathit{b}-\mathit{y}\right)}^2$ .......(ii)
Similarly, right triangle ODN
${\mathit{r}}^2$ = ${\mathit{b}}^2+{\left(\mathit{a}-\mathit{x}\right)}^2$ .......(iii)
Adding (ii) and (iii),
${\mathit{r}}^2+{\mathit{r}}^2$ = ${\mathit{a}}^2+{\mathit{b}}^2+{\left(\mathit{b}-\mathit{y}\right)}^2+{\left(\mathit{a}-\mathit{x}\right)}^2$
From (i)
$2{\mathit{r}}^2$ = ${\mathit{a}}^2+{\mathit{b}}^2+{\mathit{c}}^2$
${\mathit{r}}^2$ = $\frac{{\mathit{a}}^2+{\mathit{b}}^2+{\mathit{c}}^2}{2}$
r = $\frac{\sqrt{{\mathit{a}}^2+{\mathit{b}}^2+{\mathit{c}}^2}}{2}$