Calculate the perimeter of
In the figure, we can observe that
we have a large triangle on the outside, triangle 𝐴𝐵𝐶. And on the sides of this triangle,
we have three pairs of congruent line segments marked. Firstly, line segments 𝐶𝐸 and
𝐵𝐸 are congruent. Line segments 𝐴𝐷 and 𝐵𝐷 are
congruent. And line segments 𝐴𝐹 and 𝐶𝐹 are
congruent. And so we could describe points 𝐷,
𝐸, and 𝐹 as midpoints of their respective sides in triangle 𝐴𝐵𝐶.
Now, given that we have to find the
perimeter of 𝐷𝐸𝐶𝐹, which is the shaded polygon in the figure, then knowing that
we have the midpoints of the sides of the triangle will be very useful. Because we can apply the triangle
midsegment theorem, which is stated as the line segment connecting the midpoints of
two sides of a triangle is parallel to the third side and is half its length.
So let’s firstly consider line
segment 𝐹𝐷, which is a line connecting the midpoints of two sides of the
triangle. By the theorem, we know that it
must be parallel to the third side, which is the line segment 𝐶𝐵, and half the
length of line segment 𝐶𝐵. Since we are given on the diagram
that line segment 𝐶𝐵 has a length of 159.6 centimeters, then halving this, we can
calculate that line segment 𝐹𝐷 has a length of 79.8 centimeters.
Now let’s think about the line
segment 𝐷𝐸. By applying the theorem once more,
we know that this line segment must be parallel to the third side 𝐴𝐶 and half its
length. So we can write that 𝐷𝐸 equals
one-half 𝐴𝐶. Given the measurement of 142.4
centimeters for line segment 𝐴𝐶, then halving this gives us that the line segment
𝐷𝐸 has a length of 71.2 centimeters.
Now, we can return to the fact that
we need to find the perimeter of 𝐷𝐸𝐶𝐹, which is the distance around the outside
edge. We have worked out the length of
two of the sides of this quadrilateral, but what about the other two sides? Well, we can consider that we have
worked out that 𝐷𝐸𝐶𝐹 has two pairs of opposite sides parallel. And so, by definition, 𝐷𝐸𝐶𝐹 is
a parallelogram. And one of the properties of
parallelograms is that opposite sides are congruent. So we know that line segment 𝐶𝐸
must also be 79.8 centimeters. And line segment 𝐶𝐹 is 71.2
Therefore, to find the perimeter,
we add the lengths of the four sides of 79.8, 71.2, 79.8, and 71.2 centimeters,
which gives us an answer for the perimeter of 𝐷𝐸𝐶𝐹 as 302 centimeters.