### Video Transcript

Consider the points π΄ seven, seven; π΅ nine, negative seven; and πΆ five, one. Given that line segment π΄π· is a median of the triangle π΄π΅πΆ and π is the midpoint of this median, determine the coordinates of π· and π.

We begin by sketching the triangle π΄π΅πΆ so we can see what weβre looking for. Weβre told in the question that line segment π΄π· is a median of the triangle and that π is the midpoint of this median. And we want to determine the coordinates of π· and π. Now, we know that a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. In our case, the vertex is the point π΄, π· is the midpoint of the opposite side, πΆπ΅, and π is the midpoint of the median.

Our order of play then is to first find the point π· using the formula for the midpoint of a line segment between two points. This tells us that for two points with coordinates π₯ one, π¦ one and π₯ two, π¦ two, the midpoint of the line segment between them has coordinates π₯ one plus π₯ two over two and π¦ one plus π¦ two over two. And once weβve found our point π·, we can use this to find the midpoint π of the line segment π΄π· in the same way.

Okay, so π· is the midpoint of the line segment πΆπ΅. And so, our two points π₯ one, π¦ one and π₯ two, π¦ two are πΆ five, one and π΅ nine, negative seven. And substituting these into the formula for the midpoint, we have five plus nine over two β that is, π₯ one plus π₯ two over two, and thatβs the π₯-coordinate and π¦-coordinate one plus negative seven over two. That is 14 over two and negative six over two. So, the π· has coordinates seven, negative three. And we see that this agrees with the position of π· on our sketch.

So, we have our point π·, which is the midpoint of line segment πΆπ΅, and now we want to find the midpoint π of line segment π΄π·. So now to use our formula for π, we let π΄ be the point π₯ one, π¦ one with coordinates seven, seven and π· be the point π₯ two, π¦ two with coordinates seven, negative three. π₯ one plus π₯ two over two is seven plus seven over two. And the π¦-coordinate π¦ one plus π¦ two over two is equal to seven plus negative three over two. That is, π₯ is 14 over two and π¦ is four over two so that our midpoint π has coordinates seven, two, which again agrees with the position of π on our sketch.

The coordinates of the two points π· and π are therefore π· is seven, negative three and π is seven, two.