Question

Find the area of an equilateral triangle inscribed in the circle x2+y2 - 6x + 2y – 15 = 0.

Answer 1

Answer on Question #69437 – Math – Analytic Geometry

Question

Find the area of an equilateral triangle inscribed in circle $x^{2} + y^{2} - 6x + 2y - 15 = 0$.

Solution

Let us present the equation of the given circle

$$x^{2} + y^{2} - 6x + 2y - 15 = 0$$

in standard form

(see http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php):

$$(x^{2} - 6x + 9) + (y^{2} + 2y + 1) = 15 + 9 + 21 \Leftrightarrow (x - 3)^{2} + (y + 1)^{2} = 25.$$

Its radius $r$ can be found from the following equation:

$$r^{2} = 25 \Rightarrow r = 5.$$

Let $A$ be an area of the inscribed triangle. Then we have:

$$A = \frac{3\sqrt{3}}{4}r^{2} = \frac{3\sqrt{3}}{4} \cdot 25 = \frac{75\sqrt{3}}{4}.$$

(see https://en.wikipedia.org/wiki/Equilateral_triangle#Principal_properties).

Answer: $\frac{75\sqrt{3}}{4}$

Answer provided by https://www.AssignmentExpert.com