Use the Triangle Midsegment Theorem to divide the triangle into four triangles. Are they congruent? Can you rotate some of these triangles and get a parallelogram?
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.
Now, let's consider any triangle ABC. To make it more general, we will consider a scalene triangle.
Next, we will draw the three midsegments of the triangle which divide the triangle into four smaller triangles.
By using the Triangle Midsegment Theorem, we obtain the following three relations.
RQ = 12AB &⇒ RQ = AP
PR = 12BC &⇒ PR = CQ
PQ = 12AC &⇒ PQ = AR
From the above and by using the Side-Side-Side (SSS) Congruence Theorem, we conclude that the midsegments partitioned â–ł ABC into four congruent triangles.
Our job now is to form a parallelogram with the four triangles above. As we can see, â–ł APR, â–ł PQR, and â–ł PBQ almost form a parallelogram.
By rotating â–ł RQC around R, we can rearrange it next to â–ł APR so that the four triangles form a parallelogram.
This appears to be a parallelogram. Thanks to the Triangle Midsegment Theorem, we can be sure that the opposite sides of the quadrilateral are parallel, so the quadrilateral is in fact a parallelogram. This can be done with any triangle.
It is possible that any triangle can be partitioned into four congruent triangles that can be rearranged to form a parallelogram.
Extra
More Parallelograms
Below, we show three more rearrangements that transform the four triangles into a parallelogram.
