Video Transcript
If the coordinates 𝐴: nine, eight; 𝐵: four, negative two; and 𝐶: negative one, negative three are vertices of the triangle 𝐴𝐵𝐶, find the coordinates of the point of intersection of its medians using vectors.
We will begin by sketching a coordinate axes and plotting the three points. We know that 𝐴 has coordinates nine, eight; point 𝐵 has coordinates four, negative two; and 𝐶 has coordinates negative one, negative three. Joining the three points, we have a triangle as shown. We will call the origin 𝑂. This means that the vectors 𝐎𝐀, 𝐎𝐁, and 𝐎𝐂 are equal to nine, eight; four, negative two; and negative one, negative three, respectively.
In this question, we are asked about the medians of the triangle. A median of a triangle goes from one of the vertices to the midpoint of the opposite line segment. On our diagram, the point 𝐷 is the midpoint of 𝐵 and 𝐶. Likewise, the median from vertex 𝐵 bisects 𝐴𝐶 at point 𝐸. Finally, the median from vertex 𝐶 bisects 𝐴𝐵 at point 𝐹. These three medians intersect at a point we will call 𝑀. This point 𝑀 is known as the centroid of the triangle, and this centroid divides each of the medians in the ratio two to one. For example, the point 𝑀 divides the line segment 𝐴𝐷 in the ratio two to one such that the ratio of 𝐴𝑀 to 𝑀𝐷 is two to one. We also know that as 𝐷 is the midpoint of 𝐵𝐶, then the ratio of 𝐵𝐷 to 𝐷𝐶 is one to one.
We will now clear some space and recall the formula for finding the position vector of a point dividing a given line in a given ratio. Our formula or rule states that if 𝐴 and 𝐵 are points with position vectors lowercase 𝐚 and lowercase 𝐛 and 𝐶 divides 𝐴𝐵 in the ratio 𝜆 to 𝜇, then the position vector of 𝐶 is 𝜇 multiplied by vector 𝐚 plus 𝜆 multiplied by vector 𝐛 all divided by 𝜆 plus 𝜇. This means that the vector 𝐎𝐃 is equal to one multiplied by the vector 𝐎𝐁 plus one multiplied by the vector 𝐎𝐂 all divided by one plus one. Simplifying this, we see that vector 𝐎𝐃 is equal to vector 𝐎𝐁 plus vector 𝐎𝐂 divided by two. Whilst we could calculate this vector by substituting in our values of 𝐎𝐁 and 𝐎𝐂, at present we will leave it in this form.
Next, we can use our other ratio to calculate 𝐎𝐌. This gives us vector 𝐎𝐌 is equal to one multiplied by vector 𝐎𝐀 plus two multiplied by vector 𝐎𝐃 divided by two plus one. This can be simplified such that vector 𝐎𝐌 is equal to vector 𝐎𝐀 plus two lots of vector 𝐎𝐃 all divided by three. We have already found an expression for vector 𝐎𝐃. Substituting this into our equation for vector 𝐎𝐌, we see that vector 𝐎𝐌 is equal to vector 𝐎𝐀 plus two multiplied by vector 𝐎𝐁 plus vector 𝐎𝐂 divided by two all divided by three. The twos cancel so that the right-hand side simplifies to vector 𝐎𝐀 plus vector 𝐎𝐁 plus vector 𝐎𝐂 divided by three.
This is actually a general rule for finding the centroid of a triangle. The position vector of the centroid of a triangle is equal to one-third of the sum of the position vectors of the three vertices. We can now substitute in our values of 𝐎𝐀, 𝐎𝐁, and 𝐎𝐂. We will begin by finding the sum of the vectors nine, eight; four, negative two; and negative one, negative three. We will then need to multiply this by the scalar one-third. Adding the 𝑥-components gives us 12 as nine plus four is 13 and adding negative one is 12. The sum of the 𝑦-components is three. The vector 𝐎𝐌 is equal to one-third multiplied by the vector 12, three. This simplifies to the vector four, one as one-third of 12 is four and one-third of three is one.
As this is the vector 𝐎𝐌, we can conclude that the coordinates of the point of intersection of the medians of the triangle are four, one.
