What is length 𝑚𝐵 rounded to the nearest hundredth?
Let’s look carefully at the diagram we’ve been given. It consists of a right-angled triangle, triangle 𝐴𝐵𝐶. We’ve been told the length of the hypotenuse. It’s 10 centimeters. 𝑀 is a point somewhere inside the triangle. And we’re asked to find the length of the line 𝑀𝐵.
Let’s first consider the line 𝐵𝑌, which connects the right angle of the triangle to the opposite side. The blue lines on the two portions of 𝐴𝐶 indicate that they are both the same length. And therefore 𝐵𝑌 is a median of the triangle. Specifically, it’s the median drawn from the right angle.
A key fact which you need to remember about the medians of right-angled triangles tells us that the length of the median from the vertex of the right angle is half the length of the hypotenuse. So we can calculate the length of 𝐵𝑌. It’s 𝐴𝐶 over two. 𝐴𝐶 is 10 centimeters, and therefore 𝐵𝑌 is five centimetres. So now we know the length of 𝐵𝑌, but we want to calculate the length of 𝑀𝐵, which is just a portion of this line.
The question is how far along the length of 𝐵𝑌 is the point 𝑀. Let’s consider the other internal line in this triangle, the line 𝐴𝐿. As before, we can see that this line divides the opposite side of the triangle, in this case 𝐵𝐶, into two equal portions. And therefore 𝐴𝐿 is also a median of the triangle. How does this information help? Well 𝑀 is the point where these two medians intersect, which means it is the centroid or concurrence point of the triangle.
A key fact about the positioning of the centroid of a triangle is that it divides each median in the ratio two to one. The longer part of this ratio is always the part coming from the vertex of the triangle. So this means that the line segments 𝑀𝐵 and 𝑀𝑌 are in the ratio two to one. Or phrased another way, we can conclude the 𝑀𝐵 is two-thirds of the total length of 𝐵𝑌.
We’ve already calculated 𝐵𝑌. It’s five centimeters. And therefore we have all the information we need to now calculate the length of 𝑀𝐵. It’s two-thirds multiplied by five, which is 10 over three. Remember, the question has asked us to give this length rounded to the nearest 100th. So as a decimal, this is 3.33 centimeters.
There were two key facts that we used about the medians of triangles. Firstly, that in a right-angled triangle, the length of the median from the vertex of the right angle is always half the length of the hypotenuse. Secondly, we used the fact that the centroid of the triangle, the point where the medians intersect, divides the median in the ratio two to one.