Question Video: Applying the Triangle Midsegment Theorem to Solve a Problem Mathematics • First Year of Preparatory School

Video Transcript

In the figure shown, 𝐸, 𝐹, and 𝐷 are the midpoints of line segments 𝐵𝐶, 𝐴𝐵, and 𝐴𝐶, respectively. Find the perimeter of triangle 𝐸𝐹𝐷.

We should note that from the information we are given and the markings on the diagram, that we have three midpoints of line segments here. 𝐸, 𝐹, and 𝐷 bisect their respective line segments. So, in order to find the perimeter of triangle 𝐸𝐹𝐷, that’s the distance around the outside edge, we’ll need to calculate the lengths of the three line segments 𝐹𝐷, 𝐷𝐸, and 𝐸𝐹.

Now, because we know that there are some midpoints of lines, that might make us wonder if we could possibly apply one of the triangle midsegment theorems. We can recall that the length of the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side. Let’s look at line segment 𝐹𝐷. Line segment 𝐹𝐷 is a line segment joining the midpoints of two sides of a triangle. Therefore, its length is going to be half the length of the third side, which is line segment 𝐵𝐶. The length of 𝐵𝐶 is given as 4.6 centimeters, so half of this is 2.3 centimeters.

Now, let’s see if we can do the same to calculate the lengths of the other two sides in triangle 𝐸𝐹𝐷. We can consider the line segment 𝐷𝐸 next. Line segment 𝐷𝐸 joins the midpoints of two sides of a triangle, because it joins 𝐷, the midpoint of line segment 𝐴𝐶, and 𝐸, the midpoint of line segment 𝐵𝐶. Therefore, we know that it must be half the length of line segment 𝐴𝐵, which is the third side of the triangle. Half of 5.5 centimeters is 2.75 centimeters. And we can do the same for the length of line segment 𝐸𝐹. It joins midpoints 𝐸 and 𝐹. So, the length of 𝐸𝐹 will be half of line segment 𝐴𝐶; half of 6.2 centimeters is 3.1 centimeters.

And now to find the perimeter of triangle 𝐸𝐹𝐷, we add these three calculated lengths together. 2.3 plus 2.75 plus 3.1 is equal to 8.15 centimeters. And so, by applying the triangle midsegment theorem three times, we have determined that the perimeter of triangle 𝐸𝐹𝐷 is 8.15 centimeters.