Question

The vertices of an equilateral triangle on the coordinate plane are plotted at (-3, -√3), (0, 2√3), and (3, -√3). Which statement best describes the triangle?

A. An apothem of the triangle extends from (0, -√3) to (0, 0), and the area of the triangle is 9√3 square units.
B. An apothem of the triangle extends from (0, -√3) to (0, 3), and the area of the triangle is 973 square units.
C. An apothem of the triangle extends from (0, -√3) to (0, 3), and the area of the triangle is 2713 square units.
D. An apothem of the triangle extends from (0, -√3) to (0, 0), and the area of the triangle is 2713 square units.

Answer 1

Using the Pythagorean theorem, we can find the side length of the equilateral triangle and then calculate its area. However, the options provided for the area do not match the correct calculations, indicating a potential error in the given statements.

Step-by-step explanation:

The vertices of an equilateral triangle are given on the coordinate plane. The Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b), can be expressed as a² + b² = c². An apothem is a line segment from the center of a polygon to the midpoint of one of its sides. For an equilateral triangle on the coordinate plane, the apothem will intersect the y-axis at the triangle's centroid.

Using the given vertices, we would first calculate the side length using the distance formula derived from the Pythagorean theorem. Since the triangle is equilateral, all sides are equal. To find the area, we can use the formula for the area of an equilateral triangle: (√3/4) × side length². However, none of the given options match the correct calculations for the area of the equilateral triangle with these vertices. This suggests there may be a typo or error in the provided options or in the use of the properties of an equilateral triangle.