Since point B is a point of tangency, then \(\displaystyle{A}{B}¯¯¯¯¯≅{B}{C}¯¯¯¯¯\). Hence, \(\displaystyle∠{B}\) is a right angle and \(\displaystyle△{A}{B}{C}\) is a right triangle. Using Pythagorean Theorem,

\(\displaystyle{A}{B}^{{2}}+{B}{C}^{{2}}={A}{C}^{{2}}\)

\(\displaystyle{14}^{{2}}+{r}^{{2}}={\left({r}+{7}\right)}^{{2}}\)

\(\displaystyle{196}+{r}^{{2}}={r}^{{2}}+{14}{r}+{49}\)

196=14r+49

147=14r

10.5=r

or

r=10.5

\(\displaystyle{A}{B}^{{2}}+{B}{C}^{{2}}={A}{C}^{{2}}\)

\(\displaystyle{14}^{{2}}+{r}^{{2}}={\left({r}+{7}\right)}^{{2}}\)

\(\displaystyle{196}+{r}^{{2}}={r}^{{2}}+{14}{r}+{49}\)

196=14r+49

147=14r

10.5=r

or

r=10.5