Question

In this diagram, radius AD = 5 mm, radius BD = 12 mm and chord

CD = 8 mm. Find the exact length of AB, in surd form.

Answer 1

Answer:

The exact length of AB is $3 + 4\sqrt{10}$ milimeters.

Step-by-step explanation:

Both triangles ACD and BCD are isosceles and triangles AEC, ADE, BDE and BCE are right-angled, where E is the point where line segments AB and CD meet each other. We can determine the exact length of AB by means of two horizontal right triangles (i.e. AEC, BCE) and the Pythagorean Theorem:

$AB = \sqrt{AC^{2}-CE^{2}}+\sqrt{CB^{2}-CE^{2}}$

If we know that $AD = AC$, $BC = BD$, $AC = 5\,mm$, $BC = 12\,mm$ and $CE = 4\,mm$, then the exact length of AB is:

$AB = \sqrt{(5\,mm)^{2}-(4\,mm)^{2}}+\sqrt{(12\,mm)^{2}-(4\,mm)^{2}}$

$AB = 3 + 4\sqrt{10}\,[mm]$