Question

In a circle with center $O,$ chord $\overline{AB}$ equals chord $\overline{AC}.$ Chord $\overline{AD}$ cuts $\overline{BC}$ in $E.$ If $AC=12$ and $AE=8,$ then $AD$ equals:

$27$

$24$

$21$

$20$

$18$

Answer: E

Concepts:chord inscribed angle similarity

Difficulty rating: 1790

Small Hint:

Compare triangles $AEC$ and $ACD$

Big Hint:

Angles $ACE$ and $ADC$ subtend the equal chords $AB$ and $AC$

Solution:

Because $E$ lies on $BC$ and $AD,$ triangles $AEC$ and $ACD$ share angle $A.$ Also $\angle ACE=\angle ACB$ subtends chord $AB,$ while $\angle ADC$ subtends chord $AC.$ Since $AB=AC,$ these angles are equal. Thus $\triangle AEC\sim\triangle ACD.$ Corresponding sides give $\frac{AE}{AC}=\frac{AC}{AD}.$ Therefore $\frac{8}{12}=\frac{12}{AD},$ so $AD=18.$

Thus, the correct answer is E.

1958 AMC 12 Problem 42

Answer: E

Difficulty rating: 1790

Small Hint:

Big Hint:

Solution:

Problem 42 in Other Years