Chapter 12.8, Problem 42SPR

To determine

Find x .

Answer to Problem 42SPR

x =4

Explanation of Solution

Given:

Assume that segments that appear to be tangent are tangent.

Calculation:

We have, the product of a secant to the circle and the total length of the secant are equal :

  $\begin{array}{l}x(x+5)=3(3+9)\\ \Rightarrow x^2+5x=36\\ \Rightarrow x^2+5x-36=\mathrm{0....................}[\text{subtract 36 from each side]}\\ \Rightarrow x^2+9x-4x-36=\mathrm{0............}[\text{factor the middle term]}\\ \Rightarrow (x-4)(x+9)=0\\ \Rightarrow x-4=0\text{ or }x+9=\mathrm{0.......}[ab=0\iff a=0\text{ or }b=0]\\ \Rightarrow x=4\text{ or }x=-9\end{array}$

Since the length of the secant cannot be negative, x =4.