Video Transcript
Use the data in the figure to
determine the length of line segment 𝐷𝐹 and then the perimeter of triangle
𝐷𝐸𝐹.
Let’s begin this question then by
looking at the data that we are given. We can observe that we have a
congruent pair of line segments, 𝐴𝐹 and 𝐵𝐹. And there is another pair of
congruent line segments, 𝐴𝐸 and 𝐶𝐸. We can also note that there is a
right angle marked on the diagram. So the measure of angle 𝐴𝐷𝐶 is
90 degrees. This would also allow us to note
that the measure of angle 𝐴𝐷𝐵 would also be 90 degrees, as 𝐵𝐶 is a straight
line segment and the sum of the measures on a straight line is 180 degrees.
Now, the first length that we need
to calculate is that of the line segment 𝐷𝐹. We can consider this line segment
in the context of this triangle 𝐴𝐵𝐷, which is marked in pink. In this way, we can then see that
line segment 𝐷𝐹 is a line segment from a vertex of this triangle to the midpoint
of the opposite side. And that would indicate that line
segment 𝐷𝐹 is a median of this triangle.
We can also note that triangle
𝐴𝐵𝐷 is a right triangle. This is important because it means
we can apply the property that in a right triangle, the length of the median from
the vertex of the right angle equals half the length of the triangle’s
hypotenuse. As the median is drawn from the
vertex of the right angle, then we can write that this median 𝐷𝐹 is half the
length of the hypotenuse, which is the line segment 𝐴𝐵. From the diagram, we have that 𝐴𝐵
is 49 centimeters, and half of this would give 24.5 centimeters. And so we have found the first
required length of line segment 𝐷𝐹.
Next, we need to find the perimeter
of triangle 𝐷𝐸𝐹. And we can recall that the
perimeter is the distance around the outside edge. We have the lengths of one of the
sides, which is 𝐷𝐹. So, to work out the perimeter,
we’ll need to find the lengths of line segments 𝐸𝐹 and 𝐸𝐷.
Let’s take the line segment 𝐸𝐷
first. Just as we saw before, the line
segment 𝐸𝐷 forms a median of the right triangle 𝐴𝐶𝐷. And this median is from the vertex
containing the right angle. That means that its length will be
half the length of the hypotenuse of triangle 𝐴𝐶𝐷, which is the line segment
𝐴𝐶. We are given on the diagram that
the length of 𝐴𝐶 is 45 centimeters. So line segment 𝐸𝐷 has a length
of 22.5 centimeters.
Now, we need to find the final
length in triangle 𝐷𝐸𝐹, which is the line segment 𝐸𝐹. This time, we can’t say that line
segment 𝐸𝐹 is a median of any triangle. It is, however, a line segment
joining the midpoints of two line segments. And that brings us to one of the
triangle midsegment theorems, 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. So the length of line segment 𝐸𝐹
is half the length of the third side, which is the line segment 𝐵𝐶. And given that 𝐵𝐶 is 37
centimeters, then half of this is 18.5 centimeters.
We now have enough information to
work out the perimeter of triangle 𝐷𝐸𝐹. We can add the three side lengths
of 24.5, 22.5, and 18.5 centimeters, which gives us an answer of 65.5
centimeters. So the answers to both parts of the
question can be given as the length of line segment 𝐷𝐹 equals 24.5 centimeters and
the perimeter of triangle 𝐷𝐸𝐹 is 65.5 centimeters.
