Question

Need Help! Pretty sure the theorem used to solve is Theorem 8-5 triangle angle bisector theorem.

Answer 1

In triangle ABC, the exact length of BC =
`\(6√(6)\)` units.

In triangle ABD, we have:

`\[AB^2 = AD^2 + BD^2\]\[4^2 = AD^2 + BD^2\]\[16 = AD^2 + BD^2\]`

In triangle CBD:

`\[BC^2 = CD^2 + BD^2\]\[x^2 = (10√(2))^2 + BD^2\]\[x^2 = 200 + BD^2\]`

Now, we can substitute the value of
`\(AD^2 + BD^2\)` from the first equation into the second equation:

`\[x^2 = 200 + 16\]\[x^2 = 216\]\[x = √(216) = 6√(6)\]`

So, the exact length of BC is
`\(6√(6)\)` units.