Introduction to Determining Coordinates of an Equilateral Triangle in the Fourth Quadrant

When dealing with an equilateral triangle in coordinate geometry, it is crucial to understand how to determine the coordinates of its vertices if certain conditions are specified. This article covers determining the coordinates of point C in an equilateral triangle ABC that lies in the 4th quadrant, given points A(0, -6) and B(0, 0).

Understanding the Basic Properties of an Equilateral Triangle

An equilateral triangle has all three sides of equal length, and each interior angle is 60 degrees. Given that points A and B are (0, -6) and (0, 0), respectively, the length of segment AB is 6 units. Therefore, the length of each side of the equilateral triangle is also 6 units.

Using the Distance Formula to Determine Coordinates of C

Since vertex C is an equal distance from A and B, we can apply the distance formula to find the coordinates of C in the 4th quadrant. The distance formula is given by:

d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}

Given:

d_{1} d_{2}

Substituting the coordinates of points A and B into the distance formula, we find:

x - 0^2 (y 6)^2 x - 0^2 y - 0^2

Simplifying this equation, we get:

x^2 (y 6)^2 x^2 y^2

Solving for y:

x^2 y^2 12y 36 x^2 y^2

12y 36 0

y -3

With y known, we can find x using the angle properties of an equilateral triangle. The angle at vertex A is 60 degrees, and since point C is in the 4th quadrant, angle PAC is 30 degrees.

Using Trigonometry to Find the Exact Coordinates of C

Using the coordinates of point A and the angle 30 degrees in the 4th quadrant, we can determine the exact coordinates of C mathematically:

x can be found using the tangent function:

tan(30^circ) frac{y - (-6)}{x - 0} frac{y 6}{x}

Since (tan(30^circ) frac{sqrt{3}}{3}), we get:

frac{y 6}{x} frac{sqrt{3}}{3}

Given y -3, substituting into the equation:

frac{-3 6}{x} frac{sqrt{3}}{3}

frac{3}{x} frac{sqrt{3}}{3}

x 3sqrt{3}

Therefore, the coordinates of point C in the 4th quadrant are:

(3sqrt{3}, -3)

Conclusion and Final Verification

In conclusion, the coordinates of point C in an equilateral triangle ABC in the 4th quadrant, given points A(0, -6) and B(0, 0), are: