Chapter 13, Problem 14CUR

To determine

To calculate: The value of $x$ and $y$ .

Answer to Problem 14CUR

The value of $x$ and $y$ are $13.416$ and $8$ respectively.

Explanation of Solution

Given information: The length of tangent segment is $x$ as shown in the figure below.

  

Calculation:

It is known that if a tangent and secant is drawn from exterior point to a circle, then the square of tangent segment is equal to the product of secant segment and external secant segment.

Consider the figure as:

  

In the given figure, BA is tangent, BD is secant and BC and external secant segment. So,

  $\begin{array}{l}{\text{BA}}^2=\text{BC}\cdot \text{BD}\\ {\text{BA}}^2=\text{BC}\cdot \left(\text{BC}+\text{CD}\right)\end{array}$

To find the value of x , substitute x for BA, 9 for BC and 11 for CD in above formula:

  $\begin{array}{c}{x}^2=9\left(9+11\right)\\ x^2=9\times 20\\ x=\sqrt{180}\\ x=13.416\end{array}$

In the given figure, BA is tangent, BD is secant and BC and external secant segment. So,

  $\begin{array}{l}{\text{BA}}^2=\text{BE}\cdot \text{BF}\\ {\text{BA}}^2=\text{BE}\cdot \left(\text{BE}+\text{EF}\right)\end{array}$

To calculate the value of $y$ , substitute y for EF, 10 for BE and $\sqrt{180}$ for BA in above formula:

  $\begin{array}{c}{\left(\sqrt{180}\right)}^2=10\left(10+y\right)\\ 180=10\left(10+y\right)\\ 18=10+y\\ y=8\end{array}$

Therefore, the value of $x$ and $y$ are $13.416$ and $8$ respectively.