+0

# $\overline{BC}$ is a chord of a circle with center $O$ and area $48\pi$. Point $A$ is inside $\triangle BCO$ such

+2
36
1
+497

$\overline{BC}$ is a chord of a circle with center $O$ and area $48\pi$. Point $A$ is inside $\triangle BCO$ such that $\triangle ABC$ is equilateral and $A$ is the circumcenter of $\triangle BCO$. What is the area of triangle $ABC$?

michaelcai  Dec 5, 2017
Sort:

#1
+79819
+2

Look at the following diagram to get a feel for this :

The  larger circle will have the equation

x^2 + y^2  = 48

Since  A will be the circumcenter for BOC.....then the distance from A to O will be the same distance as from A to B  and from A to C

And since ABC is equilateral, then BAC  = 60°

Then major angle  BAC  = 300°

And using symmetry, angle OAC  = 150°

But AO = AC....so...triangle  OAC is isosceles....   and angle AOC  = 15°

And since OC is a radius of the larger circle it equals √48

So......we can find a side of the equilateral triangle - AC - thusly

OC / sin (OAC) = AC/ sin (AOC)

√48 / sin (150)  =  AC / sin (15)

AC  =  √48sin (15) / sin (150)  =

AC = √48   ( sin (45 -30) / (1/2) =

2√48 (sin45sin30 - sin30cos45) =

2√48 (√2√3 / 4   - √2 / 4)  =

√48/2 (√6 - √2 )  =

2√3 ( √6  - √2 )

√12 ( √6 - √2)  =

√72  - √24   =

6√2 - 2√6  (exact value)

Then  the equation for the circle with A as a center  passing through the vertices of BOC  is

x^2  +  (y^2 +6√2 - 2√6)^2  = ( 6√2 - 2√6)^2

And the area of equilateral triangle ABC  =  (√3 /4)s^2  =

(√3/4) (6√2 - 2√6)^2  =

(√3 / 4) ( √72 - √24)  =

(√3/ 4) ( 72  - 2 √(72*24) + 24) =

(√3/ 4) ( 96 - 2√1728)  =

(√3/ 4) (96 - 48√3)  =

12√3 ( 2 - √3)  =

24√3 - 36   units ^2   ≈  5.57  units^2

CPhill  Dec 5, 2017
edited by CPhill  Dec 5, 2017
edited by CPhill  Dec 5, 2017

### 10 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details