Answer
BD = $\sqrt 89 \approx 9.43$
Work Step by Step
In given figure 23, we can see two right triangles BCD and BCA, both right angled at C.
Given-
BC = 5, AB = 13 , AD = 4
Applying Pythagoras theorem in right triangle BCA-
$ AB^{2} $ = $ BC^{2} $ + $ AC^{2} $
$ AC^{2} $ = $ AB^{2} $ - $ BC^{2} $
$ AC^{2} $ = $ 13^{2} $ - $ 5^{2} $
$ AC^{2} $ = 169 - 25 = 144
$ AC = \sqrt (144)$ = 12
Now we can calculate DC as-
DC = AC - AD = 12 - 4 = 8
Now applying Pythagoras theorem in right triangle BCD-
$ BD^{2} $ = $ BC^{2} $ + $ DC^{2} $
$ BD^{2} $ = $ 5^{2} $ + $ 8^{2} $
$ BD^{2} $ = 25 + 64 = 89
Therefore BD = $\sqrt 89 \approx 9.43$
