Triangles

Let the side of the equilateral triangle be . The area of an equilateral triangle is given by the formula: Given the area is , we can set up the equation: Solving for : The perimeter of an equilateral triangle is , so the perimeter is: Therefore, the correct answer is 12 cm.

By the basic proportionality theorem (Thales's theorem), when a line is parallel to one side of a triangle, it divides the other two sides in the same ratio. Thus, . Thus, the correct answer is .

Since , the corresponding angles are equal. We can use the properties of similar triangles or use the method of solving linear equations to determine the value of . After solving the system of equations, we find that: Thus, the correct answer is .

The sum of the angles of a triangle is . Let the angles of the triangle be , , and . Then: Thus, the angles of the triangle are: The biggest angle is . Therefore, the correct answer is .

In the given circle, the angle subtended by the chord at the center of the circle is . Since is subtended by the same chord , we can use the property that the angle subtended at the center is twice the angle subtended at the circumference. Thus, Now, using the angle sum property in triangle , we can find: Thus, the correct answer is .

The centroid of a triangle is the average of the coordinates of its vertices. Given the vertices , we can calculate the centroid as: Therefore, the correct answer is .