The base BC of an equilateral triangle ABC lies on y- axis . The coordinates of the point C are (0,-3). If the origin is the mid-point of the base BC, find the coordinates of the points A and B.
Question
The base BC of an equilateral triangle ABC lies on y- axis . The coordinates of the point C are (0,-3). If the origin is the mid-point of the base BC, find the coordinates of the points A and B.
Solution
Given that the origin is the midpoint of the base BC, this means that the coordinates of point B are also (0,-3) because in an equilateral triangle, all sides are equal.
Now, let's find the coordinates of point A.
Since triangle ABC is equilateral, the height can be found using the formula √3/2*a, where a is the side length.
Given that the origin is the midpoint of BC, the length of BC is 2*(-3) = -6.
So, the height of the triangle is √3/2*(-6) = -3√3.
Since point A lies on the x-axis, its y-coordinate is 0.
The x-coordinate can be found by using the Pythagorean theorem.
The length of AC is the same as the length of BC, which is -6.
So, the x-coordinate of A is √((-6)^2 - (-3√3)^2) = √(36 - 27) = √9 = 3.
Therefore, the coordinates of point A are (3,0) and the coordinates of point B are (0,-3).
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