Question

Pls help
make use of cosine rule

Answer 1

Check the picture below.

let's find the internal angle for an octagon quickly

`\underset{in~degrees}{\textit{Sum of All Interior Angles}}\\\\ n\theta = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\ \theta = \stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ n=8 \end{cases}\implies 8\theta =180(8-2) \\\\\\ 8\theta =180(6)\implies 8\theta =1080\implies \theta =\cfrac{1080}{8}\implies \theta =135^o`

anyhow, we can do the same for the pentagon to get an internal angle of 108°.

Now, since the sides at CD are the same length, then we can say the pentagon's side is the same as the octagon's, and if we subtract 108° and 135° from 360°, we'd be left with 117° for ∡ACD.

`\textit{Law of Cosines}\\\\ c^2 = a^2+b^2-(2ab)\cos(C)\implies c = √(a^2+b^2-(2ab)\cos(C)) \\\\[-0.35em] ~\dotfill\\\\ AB = √(5^2+5^2~-~2(5)(5)\cos(117^o)) \implies AB = √( 50 - 50 \cos(117^o) ) \\\\\\ AB \approx √( 50 - (-22.6995) ) \implies AB \approx √( 72.6995 ) \implies AB \approx 8.53`

Make sure your calculator is in Degree mode.