Pentagon RSTUV is circumscribed about a circle Solve for x f

Pentagon RSTUV is circumscribed about a circle. Solve for x for RS = 10, ST = 13, IU = II, UV = 12, and VR = 12. The figure is not drawn to scale.

Solution

In this question we need to find x (portion between vertex R and point of touch of side RS to inscribed circle )

To solve this problem we only need to remember a basic fact about an external point and a circle. It is that the length of line segments drawn from an external point to the circle in such a way that both the line segments are tangent to the circle, are equal . ... (A)

Let the point of tangency at sides RS, ST,TU,UV and VR be A, B ,C ,D and E respectively.

Clearly, RA=x (to find).

Now , if we see RA and RE then we can observe that both orginate from extenral point R and both are tangent to inscribed circle and hence should be equal .......(i)

Thus RE=x

But VR=12 .

VR=RE+EV

=> RE+EV=12

or EV=12-x ....i

Applying (A) for point V we get

EV=VD=12-x

VD+DU=UV=12

=>DU=12-VD =12-(12-x)=x ...ii

Applying (A) at point U

we get DU=UC=x

and UC+CT=UT=11

hence CT=11-UC =11-x ---iii

Applying (A) at T we get

CT=TB=11-x

and TB+BS =TS=13

=>BS=13-TB=13-(11-x) =2+x ....iv

Finally applying (A) at S we get

BS=SA=2+x and

SA+AR=SR=10

=>AR=10-SA=10-(2+x) =8-x ---v

But AR =x (given)

thus x=8-x

=>2x=8

x=4 (answer)