In the figure above, BE is parallel to CD. What is the length of AC

**Hint** :
When parallel lines are cut by a transversal, congruent corresponding angles are formed.

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**Flashcards**

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A. monotonous | ||

B. unorthodox | ||

C. advantageous | ||

D. undeniable |

**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*.

$\frac{\mathit{\text{AC}}}{{\mathit{\text{AB}}}}=\frac{36}{12}$ | Proportion of side lengths |

$\frac{\mathit{\text{AC}}}{{9}}=\frac{36}{12}$ | Plug in AB = 9 |

$\frac{\mathit{\text{AC}}}{9}=3$ | Simplify on the right |

$\mathit{\text{AC}}=27$ | Multiply both sides by 9 |

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:**