Video Transcript
Consider the points 𝐴 seven, seven; 𝐵 nine, negative seven; and 𝐶 five, one. Given that line segment 𝐴𝐷 is a median of the triangle 𝐴𝐵𝐶 and 𝑀 is the midpoint of this median, determine the coordinates of 𝐷 and 𝑀.
We begin by sketching the triangle 𝐴𝐵𝐶 so we can see what we’re looking for. We’re told in the question that line segment 𝐴𝐷 is a median of the triangle and that 𝑀 is the midpoint of this median. And we want to determine the coordinates of 𝐷 and 𝑀. 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 𝐴, 𝐷 is the midpoint of the opposite side, 𝐶𝐵, and 𝑀 is the midpoint of the median.
Our order of play then is to first find the point 𝐷 using the formula for the midpoint of a line segment between two points. This tells us that for two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, the midpoint of the line segment between them has coordinates 𝑥 one plus 𝑥 two over two and 𝑦 one plus 𝑦 two over two. And once we’ve found our point 𝐷, we can use this to find the midpoint 𝑀 of the line segment 𝐴𝐷 in the same way.
Okay, so 𝐷 is the midpoint of the line segment 𝐶𝐵. And so, our two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two are 𝐶 five, one and 𝐵 nine, negative seven. And substituting these into the formula for the midpoint, we have five plus nine over two — that is, 𝑥 one plus 𝑥 two over two, and that’s the 𝑥-coordinate and 𝑦-coordinate one plus negative seven over two. That is 14 over two and negative six over two. So, the 𝐷 has coordinates seven, negative three. And we see that this agrees with the position of 𝐷 on our sketch.
So, we have our point 𝐷, which is the midpoint of line segment 𝐶𝐵, and now we want to find the midpoint 𝑀 of line segment 𝐴𝐷. So now to use our formula for 𝑀, we let 𝐴 be the point 𝑥 one, 𝑦 one with coordinates seven, seven and 𝐷 be the point 𝑥 two, 𝑦 two with coordinates seven, negative three. 𝑥 one plus 𝑥 two over two is seven plus seven over two. And the 𝑦-coordinate 𝑦 one plus 𝑦 two over two is equal to seven plus negative three over two. That is, 𝑥 is 14 over two and 𝑦 is four over two so that our midpoint 𝑀 has coordinates seven, two, which again agrees with the position of 𝑀 on our sketch.
The coordinates of the two points 𝐷 and 𝑀 are therefore 𝐷 is seven, negative three and 𝑀 is seven, two.
