Question

A chord of a circle divides the circle into two parts such that the squares inscribed in two parts have areas 16 and 144 square unit .the radius of the circle is

Answer 1

area of a 1st square=16
side of a square=4
area of a 2nd square =144
side of a square=12
diameter=4+12=16
radius=8

Answer 2

Radius of circle: $\bold{\sqrt{85}}$ units

Solution:

As we can see in the diagram the area of the smaller square is 16 square units which means that the sides are 4 cm each and the area of the larger square is 144 square units

OB = 12 cm, OE = X cm and OM = X+4 cm

ON = R cm,

$\begin{aligned}O N^{2} &=O M^{2}+P N^{2} \\ &=(X+4)^{2}+2^{2} \\ &=X^{2}+8 X+16+4 \\ &=X^{2}+8 X+20 \end{aligned}$

$\bold{O N^{2} = X^{2}+8 X+20=0}$ _________   (1)

Now for triangle OBC, $O C^{2}=O B^{2}+O C^{2}$

$\begin{array}{l}{R^{2}=(12-X)^{2}+6^{2}} \\ {R^{2}=144+X^{2}-24 X+36} \end{array}$

$\bold{R^{2}=X^{2}-24 X+180}$ ________ (2)

Solving equation (1) and (2) we get

$X^{2}+8 X+20=X^{2}-24 X+180$

8X + 24 X = 180 - 20

32X = 160

X=5  

Radius of circle $R^{2}=25-120+180$

$R^{2}=85}$

$\bold{R=\sqrt{85}}$ units