Question

Area of equilateral triangle in a circle x^2+y^2+2x-4y-8=0

Answer 1

Answer:

39√3 / 4

Step-by-step explanation:

First we should note that once we have the radius of the circle,

the height of our triangle is ( one radius plus half a radius ), so

height = 3 r / 2.

Also (using Pythagoras' Theorem, if necessary), in an equilateral triangle the ratio of side length to height is 2 : √3.  Therefore the base of the equilateral triangle is

base = 2 × height / √3  = √3 r.

The area of the triangle is then

A = base × height / 2

= (√3 r) × ( 3 r / 2 ) / 2

= 3√3 r² / 4.

So we just need the value of r.

Completing the squares in the equation for the circle, we get

x² + 2x + y² - 4y = 8

=>  ( x + 1 )² + ( y - 2 )² = 8 + 1² + 2² = 8 + 1 + 4 = 13

From here we can see that the radius is √13.  (Also, the centre is at (-1, 2).)

Putting r = √13 into our formula for the area of the triangle, we get

A = 3√3 × 13 / 4  =  39√3 / 4