UWorld Digital SAT®
Question of the Week

• Explanation

Correct Answer : 27

In the given figure B lies on AC and E lies on AD, so ∠CAD and ∠BAE refer to the same angle. Triangles ABE and ACD both include angle ∠CAD, and angle ∠ACD is a right angle.

It is given that BE is parallel to CD. When parallel lines are intersected by a transversal, the pairs of corresponding angles formed are congruent. Angle ∠ACD corresponds to ∠ABE, so ∠ABE is a right angle.

Triangles △ABE and △ACD have two pairs of congruent angles, so they are similar by the Angle-Angle similarity theorem. Therefore, the lengths of their corresponding sides must be proportional.

Side AC corresponds to AB, and CD corresponds to BE. Set the ratio of AC to AB equal to the ratio of CD to BE ( $\left(\frac{\mathit{\text{AC}}}{\mathit{\text{AB}}}=\frac{\mathit{\text{CD}}}{\mathit{\text{BE}}}\right)$ ). Then plug the side lengths from the given figure into the proportion.

The given length of CD is 36 and the length of BE is 12. Use the Pythagorean theorem with right triangle △ABE to find that the length of leg AB is 9. Plug this value into the proportion, and then solve for the value of AC.

The length of AC is 27.

Note: It is also possible to notice that the given lengths of leg BE (12) and hypotenuse AE (15) form a 9-12-15 Pythagorean triple to determine that the length of AB must be 9.

Things to remember:

• When parallel lines are cut by a transversal, the following pairs of angles are formed:

  
• Triangles that have two pairs of congruent angles are similar (Angle-Angle similarity theorem). Corresponding side lengths of similar triangles are proportional.