Video Transcript
Given that 𝐴𝐷 equals nine
centimeters and 𝐸𝐵 equals 𝐴𝐵, find the perimeter of triangle 𝑀𝐷𝐸.
We can start by adding in the
additional information that the length of line segment 𝐴𝐷 is nine centimeters. And we have two congruent line
segments, 𝐸𝐵 and 𝐴𝐵, so these are both 12 centimeters in length. Now, we need to find the perimeter
of triangle 𝑀𝐷𝐸, which is the total distance around the outside edge of the
triangle. That means we need to know the
lengths of the three sides, the line segments 𝐸𝐷, 𝑀𝐸, and 𝑀𝐷.
Let’s take the line segment 𝐸𝐷
first. This line segment is created
between points 𝐸 and 𝐷, which are the midpoints of line segments 𝐴𝐶 and 𝐵𝐶,
respectively. And that means that we can apply
one of the triangle midsegment theorems. The length of the line segment
joining the midpoint of two sides of a triangle is equal to half the length of the
third side. Therefore, the length of line
segment 𝐸𝐷 is half the length of the line segment 𝐴𝐵, which is the third side in
the triangle. Given that 𝐴𝐵 is 12 centimeters,
then half of this is six centimeters.
Next, we can determine the
remaining lengths of line segments 𝑀𝐸 and 𝑀𝐷. We can consider line segment 𝑀𝐸
to be part of the longer line segment 𝐸𝐵. And line segment 𝑀𝐷 is part of
the longer line segment 𝐴𝐷. These two line segments are both
medians of the largest triangle 𝐴𝐵𝐶. And their point of concurrence,
that’s the point where they meet, is at point 𝑀.
We can recall that the distance
from each vertex of a triangle to the point of concurrence of its medians is
two-thirds of the length of the median from this vertex. That would mean that the line
segment 𝑀𝐵 has a length which is two-thirds the length of the line segment
𝐸𝐵. But we’re interested in the length
of the other part of the line segment, which is line segment 𝑀𝐸. Therefore, the length of line
segment 𝑀𝐸 must be one-third the length of line segment 𝐸𝐵. And given that line segment 𝐸𝐵 is
12 centimeters, then one-third of this is four centimeters.
And we can work out the length of
line segment 𝑀𝐷 by using the same theorem. This time, we can recognize that
the line segment 𝐴𝑀 is two-thirds the length of the median of line segment
𝐴𝐷. So, the length of the line segment
𝑀𝐷 that we want to determine must be one-third the length of line segment
𝐴𝐷. Given that 𝐴𝐷 is nine centimeters
in length, then one-third of this is three centimeters.
Now, we can calculate the perimeter
of triangle 𝑀𝐷𝐸 by adding lengths of its three sides, which are six, four, and
three centimeters, which gives us the answer that the perimeter of triangle 𝑀𝐷𝐸
is 13 centimeters.
