A Mathematical Blog

The coordinates of the midpoints of the sides of a triangle are (1, 2) (0, 1) and (2, 1), find its centroid. In any triangle, medians are the three segments connecting the vertices of the triangle to the midpoints of the sides opposite to the vertices. It turns out the three medians in any triangle always intersect at a single point. That point is called the centroid of the triangle.  The question is, can we find the coordinates of the centroid of a triangle from the given coordinates of the midpoints of the sides?  Yes, we can. The definition of the centroid doesn’t really give us any clue in that direction. But there are some properties of the centroid that can help us out here.   I don’t know of any formula or rule which directly relates the coordinates of the midpoints of the sides with the coordinates of the centroid. But I do know of a relation relating the coordinates of the centroid with the coordinates of the vertices of a triangle. The x and y...

I will begin by clarifying the meaning of ‘ angle subtended by an arc at a point ’.  An arc is any portion of the circumference(boundary) of the circle. Connecting the endpoints of an arc with the center of the circle results in the formation of the central angle by the arc . This is what we are referring to as the angle subtended by an arc . Particularly, this is the angle subtended by an arc at the center of the circle, hence the name central angle .  An important theorem on circles states that the central angle of an arc is twice the measure of the angle the arc subtends at any point on the remaining portion of the circumference of the circle. ‘Remaining portion of the circumference’ is just anywhere on the boundary of the circle except the arc in question. In the figure, ABC is an arc, and D is a point on the (rest of the) circumference of the circle. So if x is the angle subtended by arc ABC at (any) point D on the (rest of the) circumference, as per the theor...