SAT Mathematics Practice Test 5

Triangles ADB and CDB are congruent to each other because they are both $30°-60°-90°$ triangles and share the side $\stackrel{-}{\mathit{B}\mathit{D}}.$ In triangle ADB, side $\stackrel{-}{\mathit{A}\mathit{D}}$ is opposite to the angle $30°$; therefore, the length of $\stackrel{-}{\mathit{A}\mathit{D}}$ is half the length of hypotenuse $\stackrel{-}{\mathit{A}\mathit{B}}$ . Since the triangles are congruent, $\mathit{A}\mathit{B}=\mathit{B}\mathit{C}=12.$ So the length of $\stackrel{-}{\mathit{A}\mathit{D}}$ is $\frac{12}{2}=6$
Choice A is incorrect. If the length of $\stackrel{-}{\mathit{A}\mathit{D}}$ were 4, then the length of $\stackrel{-}{\mathit{A}\mathit{B}}$ would be 8. However, this is incorrect because $\stackrel{-}{\mathit{A}\mathit{B}}$ is congruent to $\stackrel{-}{\mathit{B}\mathit{C}}$,which has a length of $12$. Choices C and D are also incorrect. Following the same procedures as used to test choice A gives $\stackrel{-}{\mathit{A}\mathit{B}}$ a length of $12\sqrt{2}$ for choice C and $12\sqrt{3}$ for choice D. However, these results cannot be true because $\stackrel{-}{\mathit{A}\mathit{B}}$ is congruent to $\stackrel{-}{\mathit{B}\mathit{C}}$,which has a length of $12$.