The length of BC is [tex]4 \sqrt{5}[/tex].
Solution:
Given ABC is a right triangle.
AC is the hypotenuse and BD is the altitude.
AB and BC are legs of the triangle ABC.
AC = 16 and DC = 5
Leg rule of geometric mean theorem:
[tex]$\frac{\text { hypotenuse }}{\text { leg }}=\frac{\text { leg }}{\text { part }}$[/tex]
[tex]$\Rightarrow \frac{AC}{BC}=\frac{BC}{DC}$[/tex]
[tex]$\Rightarrow \frac{16}{x}=\frac{x}{5}$[/tex]
Do cross multiplication.
[tex]\Rightarrow 16\times 5 = x\times x[/tex]
[tex]\Rightarrow 80= x^2[/tex]
[tex]\Rightarrow 16\times 5= x^2[/tex]
Taking square root on both sides.
[tex]\Rightarrow \sqrt{16\times 5} = \sqrt{x^2}[/tex]
[tex]\Rightarrow \sqrt{4^2\times 5} = \sqrt{x^2}[/tex]
square and square roots get canceled, we get
[tex]\Rightarrow 4\sqrt{ 5} = x[/tex]
The length of BC is [tex]4 \sqrt{5}[/tex].
