+0

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

+2
663
1
+598

$\overline{BC}$ 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 $ABC$?

Dec 5, 2017

### 1+0 Answers

#1
+94295
+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

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

### New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.