SAT Mathematics Practice Test 5

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