Question

Part 311Use the relationships in circle O to find the missing measures in circle S.1 pointIf WR = 12 units and MN = 8 units, determine MW.Type your answerwMRNePrestipus

Answer 1

From the properties of secant line and the tangent to the circle

If a secant segment and tangent segment are drawn to a circle from the same external point, the length of the tangent segment is the geometric mean between the length of the secant segment and the length of the external part of the secant segment.

`\frac{Whole\text{ Secant}}{\tan gent\text{ Line}}=\frac{\tan gent}{External\text{ Secant Part}}`

In the given figure we hvae :

Whole secant length (RM),

Tangnt line MN = 8 units

External secant part (WM)

Since RM = WR + WM

RM=12 + WM

Susbtitute the value:

`\begin{gathered} \frac{Whole\text{ Secant}}{\tan gent\text{ Line}}=\frac{\tan gent}{External\text{ Secant Part}} \\ (RM)/(MN)=(MN)/(MW) \\ (12+MW)/(8)=(8)/(MW) \\ \text{Apply crossmultiplication:} \\ MW(12+MW)=8*8 \\ 12MW+(MW)^2=64 \\ \text{ Let MW = x} \\ 12x+x^2=64 \\ x^2+12x-64=0 \\ \text{ Factorize:} \\ x^2+16x-4x-64=0 \\ x(x+16)-4(x+16)=0 \\ (x-4)(x+16)=0 \\ \text{ So, x = 4, -16} \\ \text{ Since measurement cannot be negative thus: x = 4 unit} \\ x\text{ = }MW=4 \end{gathered}`

Answer : MW= 4 units