Quadrilateral ABCD is inscribed in a circle with ∠ BAC = 70°, ∠ ADB = 40°, AD = 4, and BC = 6. What is AC?

2015 AMC 12B Problem 13

Attempt Problem 13 of the 2015 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

Quadrilateral

ABCD

is inscribed in a circle with

\angle BAC = 70^\circ,

\angle ADB = 40^\circ,

AD = 4,

and

BC = 6.

What is

AC?

3 + \sqrt5

6

\dfrac92\sqrt2

8 - \sqrt2

7

\angle BAC

and

\angle BDC

subtend the same arc

BC

Find

\angle ABC,

then compare it with

\angle BAC

to show

\triangle ABC

is isosceles

Angles

BAC

and

BDC

subtend arc

BC,

\angle BDC = 70^\circ.

Then

\angle ADC = \angle ADB

+ \angle BDC = 110^\circ.

Since

ABCD

is cyclic,

\angle ABC = 180^\circ - 110^\circ = 70^\circ

= \angle BAC.

Thus

\triangle ABC

is isosceles with

AC = BC = 6.

Thus, the correct answer is B.