The triangle ABC is an equilateral triangle. This means that each angle equals 60°. Hence, the angle at B is 60°.
The length of each side of ABC is given to be 3 cm long.
We can get the length of side AD by solving the triangle ABD using the Cosine Rule given to be:
[tex]a^2=b^2+c^2-2bc\cos A[/tex]
Since we're considering triangle ABD, and we have the measure of angle B, we can use the relationship:
[tex]b^2=a^2+d^2-2ad\cos B[/tex]
Note that a, b, and d are the sides, such that:
[tex]\begin{gathered} a=BD=BC+CD=3+4=7\operatorname{cm} \\ b=AD \\ d=AB=3\operatorname{cm} \end{gathered}[/tex]
Substituting these values, we have:
[tex]\begin{gathered} AD^2=7^2+3^2-2(7\times3\times\cos 60) \\ AD^2=49+9-42\cos 60 \\ AD^2=37 \\ AD=\sqrt[]{37} \\ AD=6.08\operatorname{cm} \end{gathered}[/tex]
The length of AD is 6.08 cm to 2 decimal places.
